Trig limit using pythagorean identity

WEBVTT
Kind: captions
Language: en

00:00:00.305 --> 00:00:01.846
- [Voiceover] Let's see
if we can find the limit

00:00:01.846 --> 00:00:04.932
as theta approaches zero
of one minus cosine theta

00:00:04.932 --> 00:00:07.299
over two sine squared theta.

00:00:07.299 --> 00:00:08.673
And like always, pause
the video and see if you

00:00:08.673 --> 00:00:10.646
could work through this.

00:00:10.646 --> 00:00:12.516
Alright, well our first
temptation is to say,

00:00:12.516 --> 00:00:14.449
"Well, this is going to be the same thing

00:00:14.449 --> 00:00:17.699
"as the limit of one minus cosine theta

00:00:19.647 --> 00:00:21.599
"as X approaches, or not X,

00:00:21.599 --> 00:00:23.766
"as theta approaches zero.

00:00:24.644 --> 00:00:27.331
"of theta, as theta approaches zero,

00:00:27.331 --> 00:00:28.664
"over the limit,

00:00:30.159 --> 00:00:32.842
"as theta approaches zero

00:00:32.842 --> 00:00:35.175
"of two sine squared theta."

00:00:38.126 --> 00:00:40.275
Now, both of these expressions

00:00:40.275 --> 00:00:41.761
which could be used to define a function,

00:00:41.761 --> 00:00:43.436
that they'd be continuous
if you graph them.

00:00:43.436 --> 00:00:46.212
They'd be continuous at theta equals zero,

00:00:46.212 --> 00:00:47.717
so the limit is going
to be the same thing,

00:00:47.717 --> 00:00:50.585
as just evaluating them
at theta equals zero

00:00:50.585 --> 00:00:55.393
So this is going to be equal
to one minus cosine of zero

00:00:55.393 --> 00:00:57.893
over two sine squared of zero.

00:01:00.320 --> 00:01:04.487
Now, cosine of zero is one and
then one minus one is zero,

00:01:05.773 --> 00:01:08.986
and sine of zero is
zero, and you square it,

00:01:08.986 --> 00:01:10.605
You still got zero and
you multiply times two,

00:01:10.605 --> 00:01:11.438
you still got zero.

00:01:11.438 --> 00:01:13.812
So you got zero over zero.

00:01:13.812 --> 00:01:16.910
So once again, we have
that indeterminate form.

00:01:16.910 --> 00:01:18.398
And once again, this indeterminate form

00:01:18.398 --> 00:01:21.105
when you have zero over zero,
doesn't mean to give up,

00:01:21.105 --> 00:01:22.863
it doesn't mean that
the limit doesn't exist.

00:01:22.863 --> 00:01:24.174
It just means, well maybe

00:01:24.174 --> 00:01:26.121
there's some other
approaches here to work on.

00:01:26.121 --> 00:01:28.909
If you got some non-zero
number divided by zero,

00:01:28.909 --> 00:01:30.731
then you say, okay that
limit doesn't exist

00:01:30.731 --> 00:01:33.011
and you would say, well you
just say it doesn't exist.

00:01:33.011 --> 00:01:35.689
But let's see what we can
do to maybe, to maybe think

00:01:35.689 --> 00:01:38.703
about this expression in a different way.

00:01:38.703 --> 00:01:39.870
So if we said,

00:01:40.725 --> 00:01:41.558
so let's just say that this,

00:01:41.558 --> 00:01:44.342
let me use some other colors here.

00:01:44.342 --> 00:01:46.068
Let's say that this right over here

00:01:46.068 --> 00:01:46.901
is F of X.

00:01:48.067 --> 00:01:51.817
So, F of X is equal to
one minus cosine theta

00:01:53.652 --> 00:01:55.985
over two sine squared theta,

00:01:57.313 --> 00:01:59.305
and let's see if we can
rewrite it in some way

00:01:59.305 --> 00:02:02.087
that at least the limit
as theta approaches zero

00:02:02.087 --> 00:02:03.877
isn't going to, we're
not gonna get the same

00:02:03.877 --> 00:02:05.814
zero over zero.

00:02:05.814 --> 00:02:08.058
Well, we can, we got
some trig functions here,

00:02:08.058 --> 00:02:09.779
so maybe we can use some
of our trig identities

00:02:09.779 --> 00:02:11.249
to simplify this.

00:02:11.249 --> 00:02:12.848
And the one that jumps out at me

00:02:12.848 --> 00:02:15.894
is that we have the sine squared of theta

00:02:15.894 --> 00:02:19.552
and we know from the
Pythagorean, Pythagorean Identity

00:02:19.552 --> 00:02:21.392
in Trigonometry, it comes
straight out of the unit circle

00:02:21.392 --> 00:02:22.988
definition of sine and cosine.

00:02:22.988 --> 00:02:26.957
We know that, we know
that sine squared theta

00:02:26.957 --> 00:02:30.374
plus cosine squared theta is equal to one

00:02:31.494 --> 00:02:34.045
or, we know that sine squared theta

00:02:34.045 --> 00:02:36.577
is one minus cosine squared theta.

00:02:36.577 --> 00:02:38.863
One minus cosine squared theta.

00:02:38.863 --> 00:02:40.991
So we could rewrite this.

00:02:40.991 --> 00:02:44.241
This is equal to one minus cosine theta

00:02:45.770 --> 00:02:49.603
over two times one minus
cosine squared theta.

00:02:53.655 --> 00:02:56.014
Now, this is one minus cosine theta.

00:02:56.014 --> 00:02:58.826
This is a one minus cosine squared theta,

00:02:58.826 --> 00:03:00.753
so it's not completely obvious yet

00:03:00.753 --> 00:03:02.588
of how you can simplify it,

00:03:02.588 --> 00:03:04.164
until you realize that
this could be viewed

00:03:04.164 --> 00:03:05.862
as a difference of squares.

00:03:05.862 --> 00:03:07.798
If you view this as,

00:03:07.798 --> 00:03:12.361
if you view this as A
squared minus B squared,

00:03:12.361 --> 00:03:14.845
we know that this can
be factored as A plus B

00:03:14.845 --> 00:03:16.485
times A minus B.

00:03:16.485 --> 00:03:18.250
So I could rewrite this.

00:03:18.250 --> 00:03:21.938
This is equal to one minus cosine theta

00:03:21.938 --> 00:03:23.188
over two times,

00:03:25.612 --> 00:03:28.556
I could write this as
one plus cosine theta

00:03:28.556 --> 00:03:30.598
times one minus cosine theta.

00:03:30.598 --> 00:03:34.598
One plus cosine theta
times one minus, one minus

00:03:37.337 --> 00:03:38.593
cosine theta.

00:03:38.593 --> 00:03:40.318
And now this is interesting.

00:03:40.318 --> 00:03:41.703
I have one minus cosine theta

00:03:41.703 --> 00:03:43.732
in the numerator and I have
a one minus cosine theta

00:03:43.732 --> 00:03:45.164
in the denominator.

00:03:45.164 --> 00:03:46.415
Now we might be tempted to say,

00:03:46.415 --> 00:03:49.045
"Oh, let's just cross that out with that

00:03:49.045 --> 00:03:51.641
"and we would get, we would simplify it

00:03:51.641 --> 00:03:54.641
"and get F of X is equal to one over

00:03:55.496 --> 00:03:57.776
"and we could distribute this two now."

00:03:57.776 --> 00:04:01.264
We could say, "Two plus two cosine theta."

00:04:01.264 --> 00:04:02.097
We could say,

00:04:02.097 --> 00:04:03.338
"Well, aren't these the same thing?"

00:04:03.338 --> 00:04:05.151
And we would be almost right,

00:04:05.151 --> 00:04:07.852
because F of X, this one right over here,

00:04:07.852 --> 00:04:10.218
this, this is defined

00:04:10.218 --> 00:04:12.234
this right over here is defined

00:04:12.234 --> 00:04:14.567
when theta is equal to zero,

00:04:15.572 --> 00:04:17.473
while this one is not defined when theta

00:04:17.473 --> 00:04:18.306
is equal to zero.

00:04:18.306 --> 00:04:19.464
When theta is equal to zero,

00:04:19.464 --> 00:04:21.637
you have a zero in the denominator.

00:04:21.637 --> 00:04:23.591
And so what we need to do in order

00:04:23.591 --> 00:04:25.435
for this F of X or in order

00:04:25.435 --> 00:04:27.715
to be, for this to be the same thing,

00:04:27.715 --> 00:04:31.548
we have to say, theta
cannot be equal to zero.

00:04:32.807 --> 00:04:34.602
But now let's think about the limit again.

00:04:34.602 --> 00:04:37.216
Essentially, what we want
to do is we want to find

00:04:37.216 --> 00:04:40.566
the limit as theta approaches zero

00:04:40.566 --> 00:04:41.399
of F of X.

00:04:43.683 --> 00:04:46.707
And we can't just do direct substitution

00:04:46.707 --> 00:04:49.521
into, if we do, if we
really take this seriously,

00:04:49.521 --> 00:04:50.354
'cause we're gonna like,

00:04:50.354 --> 00:04:51.887
"Oh well, if I try to put zero here,

00:04:51.887 --> 00:04:53.177
"it says theta cannot be equal to zero

00:04:53.177 --> 00:04:55.110
"F of X is not defined at zero."

00:04:55.110 --> 00:04:56.644
This expression is defined at zero

00:04:56.644 --> 00:04:57.477
but this tells me,

00:04:57.477 --> 00:05:00.091
"Well, I really shouldn't
apply zero to this function."

00:05:00.091 --> 00:05:02.460
But we know that if we
can find another function

00:05:02.460 --> 00:05:04.946
that is defined, that is
the exact same thing as

00:05:04.946 --> 00:05:07.123
F of X except at zero,

00:05:07.123 --> 00:05:09.805
and it is continuous at zero.

00:05:09.805 --> 00:05:11.258
And so we could say,

00:05:11.258 --> 00:05:14.675
"G of X is equal to one over two plus two

00:05:16.959 --> 00:05:18.876
"cosine theta."

00:05:18.876 --> 00:05:20.441
Well then we know this
limit is going to be the

00:05:20.441 --> 00:05:22.739
exact same thing as the limit

00:05:22.739 --> 00:05:25.739
of G of X, as theta approaches zero.

00:05:27.364 --> 00:05:29.187
Once again, these two functions

00:05:29.187 --> 00:05:31.899
are identical except F of X is not defined

00:05:31.899 --> 00:05:33.344
at theta equals zero,

00:05:33.344 --> 00:05:35.031
while G of X is.

00:05:35.031 --> 00:05:36.739
But the limits as theta approaches zero

00:05:36.739 --> 00:05:38.962
are going to be the same.

00:05:38.962 --> 00:05:41.423
And we've seen that in previous videos.

00:05:41.423 --> 00:05:42.758
And I know what a lot of you are thinking.

00:05:42.758 --> 00:05:44.518
Sal, this seems like a very, you know,

00:05:44.518 --> 00:05:46.507
why don't I just, you
know, do this algebra here.

00:05:46.507 --> 00:05:47.988
Cross these things out of this.

00:05:47.988 --> 00:05:50.051
Substitute zero for theta.

00:05:50.051 --> 00:05:52.101
Well you could do that and
you would get the answer,

00:05:52.101 --> 00:05:54.872
but you need to be clear
if, or it's important

00:05:54.872 --> 00:05:57.163
to be mathematically clear
of what you are doing.

00:05:57.163 --> 00:06:00.352
If you do that, if you
just crossed these two out

00:06:00.352 --> 00:06:01.813
and all of a sudden you're expression

00:06:01.813 --> 00:06:03.813
becomes defined at zero,

00:06:04.931 --> 00:06:07.217
you are now dealing with
a different expression

00:06:07.217 --> 00:06:09.532
or a different function definition.

00:06:09.532 --> 00:06:11.698
So to be clear, if you want
to say this is the function

00:06:11.698 --> 00:06:12.940
you're finding the limit of,

00:06:12.940 --> 00:06:14.527
you have to put this constraint in

00:06:14.527 --> 00:06:16.381
to make sure it has the exact same domain.

00:06:16.381 --> 00:06:18.504
But lucky for us, we can say,

00:06:18.504 --> 00:06:21.288
if we've had another, another
function that's continuous

00:06:21.288 --> 00:06:24.203
at that point that doesn't
have that gap there,

00:06:24.203 --> 00:06:26.385
that doesn't have that
point discontinuity out,

00:06:26.385 --> 00:06:28.399
the limits are going to be equivalent.

00:06:28.399 --> 00:06:31.016
So the limit as theta
approaches zero of G of X,

00:06:31.016 --> 00:06:32.306
well, that's just going to be

00:06:32.306 --> 00:06:34.594
since it's continuous at zero.

00:06:34.594 --> 00:06:35.913
We could say that's just going to be,

00:06:35.913 --> 00:06:36.915
we can just substitute.

00:06:36.915 --> 00:06:39.770
That's going to be equal to G of zero

00:06:39.770 --> 00:06:43.147
which is equal to one over two plus

00:06:43.147 --> 00:06:46.980
cosine two, one over two
plus two times cosine

00:06:47.906 --> 00:06:49.096
of zero.

00:06:49.096 --> 00:06:50.605
Cosine of zero is one,

00:06:50.605 --> 00:06:52.364
so it's just one over two plus two,

00:06:52.364 --> 00:06:54.561
which is equal to,

00:06:54.561 --> 00:06:56.408
deserve a little bit of a drum roll here.

00:06:56.408 --> 00:06:58.759
Which is equal to one fourth.

00:06:58.759 --> 00:07:00.092
And we are done.


Limits by rationalizing

WEBVTT
Kind: captions
Language: en

00:00:00.461 --> 00:00:01.744
- [Voiceover] Let's see
if we can find the limit

00:00:01.744 --> 00:00:03.261
as x approaches negative 1

00:00:03.261 --> 00:00:04.642
of x plus 1

00:00:04.642 --> 00:00:08.748
over the square root of x plus 5 minus 2.

00:00:08.748 --> 00:00:10.444
So, our first reaction might just be,

00:00:10.444 --> 00:00:12.163
okay, well let's just
use our limit properties

00:00:12.163 --> 00:00:12.996
a little bit,

00:00:12.996 --> 00:00:15.466
this is going to be the
same thing as the limit

00:00:15.466 --> 00:00:19.133
as x approaches negative 1 of x plus 1

00:00:20.351 --> 00:00:21.184
over,

00:00:22.029 --> 00:00:23.279
over the limit,

00:00:25.972 --> 00:00:27.316
the limit

00:00:27.316 --> 00:00:29.808
as x approaches negative 1

00:00:29.808 --> 00:00:32.975
of square root of x plus 5 minus 2.

00:00:35.217 --> 00:00:36.782
And then we could say, all right,

00:00:36.782 --> 00:00:38.334
this thing up here,

00:00:38.334 --> 00:00:39.819
x plus 1, if this is,

00:00:39.819 --> 00:00:42.786
if we think about the
graph y equals x plus 1,

00:00:42.786 --> 00:00:44.967
it's continuous everywhere, especially at

00:00:44.967 --> 00:00:47.685
x equals negative 1, and
so to evaluate this limit,

00:00:47.685 --> 00:00:49.646
we just have to evaluate
this expression at

00:00:49.646 --> 00:00:51.005
x equals negative 1,

00:00:51.005 --> 00:00:54.877
so this numerator's just going
to evaluate to negative 1

00:00:54.877 --> 00:00:55.710
plus 1.

00:00:56.622 --> 00:00:58.316
And then our denominator,

00:00:58.316 --> 00:01:01.578
square root of x plus 5 minus
2 isn't continuous everywhere

00:01:01.578 --> 00:01:04.698
but it is continuous
at x equals negative 1

00:01:04.698 --> 00:01:06.065
and so we can do the same thing.

00:01:06.065 --> 00:01:08.679
We can just substitute negative 1 for x,

00:01:08.679 --> 00:01:11.765
so this is going to be the
square root of negative 1

00:01:11.765 --> 00:01:13.015
plus 5 minus 2.

00:01:14.591 --> 00:01:16.397
Now, what does this evaluate to?

00:01:16.397 --> 00:01:19.062
Well, in the numerator we get a zero,

00:01:19.062 --> 00:01:21.715
and in the denominator,
negative 1 plus 5 is 4,

00:01:21.715 --> 00:01:23.873
take the principle root is 2, minus 2,

00:01:23.873 --> 00:01:25.552
we get zero again,

00:01:25.552 --> 00:01:26.704
so we get,

00:01:26.704 --> 00:01:27.621
we got zero

00:01:29.205 --> 00:01:30.038
over zero.

00:01:30.972 --> 00:01:33.317
Now, when you see that, you
might be tempted to give up.

00:01:33.317 --> 00:01:35.906
You say, oh, look, there's
a zero in the denominator,

00:01:35.906 --> 00:01:37.480
maybe this limit doesn't exist,

00:01:37.480 --> 00:01:39.017
maybe I'm done here, what do I do?

00:01:39.017 --> 00:01:43.079
And if this was non-zero
up here in the numerator,

00:01:43.079 --> 00:01:45.854
if you're taking a non-zero
value and dividing it by zero,

00:01:45.854 --> 00:01:46.904
that is undefined

00:01:46.904 --> 00:01:48.783
and your limit would not exist.

00:01:48.783 --> 00:01:51.056
But when you have zero over zero,

00:01:51.056 --> 00:01:52.482
this is indeterminate form,

00:01:52.482 --> 00:01:55.897
it doesn't mean necessarily
that your limit does not exist,

00:01:55.897 --> 00:01:57.906
and as we'll see in this
video and many future ones,

00:01:57.906 --> 00:02:01.837
there are tools at our
disposal to address this,

00:02:01.837 --> 00:02:03.772
and we will look at one of them.

00:02:03.772 --> 00:02:05.495
Now, the tool that we're going to look at

00:02:05.495 --> 00:02:08.483
is is there another way of
rewriting this expression

00:02:08.483 --> 00:02:11.256
so we can evaluate its limit

00:02:11.256 --> 00:02:14.080
without getting the zero over zero?

00:02:14.080 --> 00:02:16.014
Well, let's just rewrite,

00:02:16.014 --> 00:02:18.979
let's just take this and give it,

00:02:18.979 --> 00:02:21.226
so let's take this thing right over here,

00:02:21.226 --> 00:02:23.536
and let's say this is g of x,

00:02:23.536 --> 00:02:25.364
so essentially what we're
trying to do is find the limit

00:02:25.364 --> 00:02:26.197
of g of x

00:02:26.197 --> 00:02:28.183
as x approaches negative 1,

00:02:28.183 --> 00:02:31.994
so we can write g of x is equal to

00:02:31.994 --> 00:02:34.505
x plus 1 and the only reason
I'm defining it as g of x

00:02:34.505 --> 00:02:37.250
is just to be able to think of
it more clearly as a function

00:02:37.250 --> 00:02:38.698
and manipulate the function

00:02:38.698 --> 00:02:40.118
and then think about similar functions,

00:02:40.118 --> 00:02:42.201
over x plus 5 minus 2,

00:02:44.438 --> 00:02:47.734
or x plus 1 over the square
root of x plus 5 minus 2.

00:02:47.734 --> 00:02:49.762
Now, the technique we're
going to use is when you get

00:02:49.762 --> 00:02:51.602
this indeterminate form and
if you have a square root

00:02:51.602 --> 00:02:53.526
in either the numerator
or the denominator,

00:02:53.526 --> 00:02:55.986
it might help to get
rid of that square root

00:02:55.986 --> 00:02:58.787
and this is often called
rationalizing the expression.

00:02:58.787 --> 00:03:00.989
In this case, you have a
square root in the denominator,

00:03:00.989 --> 00:03:02.562
so it would be rationalizing

00:03:02.562 --> 00:03:04.643
the denominator,

00:03:04.643 --> 00:03:06.823
and so, this would be,

00:03:06.823 --> 00:03:07.848
the way we would do it

00:03:07.848 --> 00:03:09.822
is we would be leveraging
our knowledge of difference

00:03:09.822 --> 00:03:10.695
of squares.

00:03:10.695 --> 00:03:11.593
We know,

00:03:11.593 --> 00:03:13.838
we know that a plus b

00:03:13.838 --> 00:03:15.903
times a minus b

00:03:15.903 --> 00:03:18.200
is equal to a squared minus b squared,

00:03:18.200 --> 00:03:19.937
you learned that in algebra

00:03:19.937 --> 00:03:21.447
a little while ago,

00:03:21.447 --> 00:03:25.061
or if we had the square root of a plus b

00:03:25.061 --> 00:03:27.641
and we were to multiply that
times the square root of a

00:03:27.641 --> 00:03:30.505
minus b, well that would be
the square root of a squared

00:03:30.505 --> 00:03:31.844
which is just going to be a,

00:03:31.844 --> 00:03:33.740
minus b squared,

00:03:33.740 --> 00:03:36.689
so we can just leverage these ideas

00:03:36.689 --> 00:03:38.772
to get rid of this radical down here.

00:03:38.772 --> 00:03:40.344
The way we're going to do it

00:03:40.344 --> 00:03:42.377
is we're going to multiply the
numerator and the denominator

00:03:42.377 --> 00:03:44.495
by the square root of x plus 5

00:03:44.495 --> 00:03:45.662
plus 2, right?

00:03:46.879 --> 00:03:48.098
We have the minus 2

00:03:48.098 --> 00:03:50.058
so we multiply it times the plus 2,

00:03:50.058 --> 00:03:51.921
so let's do that.

00:03:51.921 --> 00:03:53.380
So we have

00:03:53.380 --> 00:03:56.130
square root of x plus 5 plus 2

00:03:57.932 --> 00:03:59.924
and we're going to multiply
the numerator times

00:03:59.924 --> 00:04:01.694
the same thing, 'cause we
don't want to change the value

00:04:01.694 --> 00:04:02.548
of the expression.

00:04:02.548 --> 00:04:03.824
This is 1.

00:04:03.824 --> 00:04:06.458
So, if we take the expression
divided by the same expression

00:04:06.458 --> 00:04:07.721
it's going to be 1,

00:04:07.721 --> 00:04:08.788
so this is,

00:04:08.788 --> 00:04:11.260
so square root of x plus 5 plus 2

00:04:11.260 --> 00:04:12.850
and so this is going to be equal to,

00:04:12.850 --> 00:04:16.183
this is going to be equal to x plus 1

00:04:17.401 --> 00:04:19.234
times the square root,

00:04:20.269 --> 00:04:23.408
times the square root of x plus 5

00:04:23.408 --> 00:04:24.241
plus 2

00:04:25.097 --> 00:04:27.760
and then the denominator is going to be,

00:04:27.760 --> 00:04:28.985
well, it's going to be x,

00:04:28.985 --> 00:04:30.416
the square root of x plus 5 squared

00:04:30.416 --> 00:04:32.999
which would be just x plus 5

00:04:35.199 --> 00:04:37.322
and then minus 2 squared,

00:04:37.322 --> 00:04:38.826
minus 4,

00:04:38.826 --> 00:04:43.403
and so this down here
simplifies to x plus 5 minus 4

00:04:43.403 --> 00:04:44.980
is just x plus 1

00:04:44.980 --> 00:04:46.313
so this is just,

00:04:47.474 --> 00:04:49.474
this is just x plus 1

00:04:50.639 --> 00:04:52.703
and it probably jumps out at
you that both the numerator

00:04:52.703 --> 00:04:54.951
and the denominator
have an x plus 1 in it,

00:04:54.951 --> 00:04:57.000
so maybe we can simplify,

00:04:57.000 --> 00:04:58.594
so we can simplify by just say,

00:04:58.594 --> 00:05:02.704
well, g of x is equal to
the square root of x plus 5

00:05:02.704 --> 00:05:03.537
plus 2.

00:05:04.585 --> 00:05:06.687
Now, some of you might be
feeling a little off here,

00:05:06.687 --> 00:05:07.780
and you would be correct.

00:05:07.780 --> 00:05:09.735
Your spider senses would be,

00:05:09.735 --> 00:05:10.568
is this,

00:05:11.422 --> 00:05:13.564
is this definitely the same thing

00:05:13.564 --> 00:05:16.299
as what we originally had
before we cancelled out

00:05:16.299 --> 00:05:17.700
the x plus 1s?

00:05:17.700 --> 00:05:19.346
And the answer is the way I just wrote it

00:05:19.346 --> 00:05:21.624
is not the exact same thing.

00:05:21.624 --> 00:05:24.674
It is the exact same
thing everywhere except at

00:05:24.674 --> 00:05:26.481
x equals negative 1.

00:05:26.481 --> 00:05:28.367
This thing right over here is defined

00:05:28.367 --> 00:05:30.038
at x equals negative 1.

00:05:30.038 --> 00:05:31.940
This thing right over here is not defined

00:05:31.940 --> 00:05:33.503
at x equals negative 1,

00:05:33.503 --> 00:05:35.050
and g of x was not,

00:05:35.050 --> 00:05:35.944
was not,

00:05:35.944 --> 00:05:37.126
so g of x right over here,

00:05:37.126 --> 00:05:41.239
you don't get a good result
when you try x equals negative 1

00:05:41.239 --> 00:05:43.869
and so in order for this
to truly be the same thing

00:05:43.869 --> 00:05:44.886
as g of x,

00:05:44.886 --> 00:05:46.905
the same function, we have to say

00:05:46.905 --> 00:05:49.655
for x not equal to negative 1.

00:05:51.522 --> 00:05:54.682
Now, this is a simplified
version of g of x.

00:05:54.682 --> 00:05:55.679
It is the same thing.

00:05:55.679 --> 00:05:57.012
For any input x,

00:05:58.167 --> 00:06:01.136
that g of x is defined, this
is going to give you the same

00:06:01.136 --> 00:06:03.942
output, and this is the
exact same domain now,

00:06:03.942 --> 00:06:05.504
now that we've put this constraint in,

00:06:05.504 --> 00:06:06.337
as g of x.

00:06:08.616 --> 00:06:11.007
Now you might say, okay,
well how does this help us?

00:06:11.007 --> 00:06:14.451
Because we want to find the
limit as x approaches negative 1

00:06:14.451 --> 00:06:17.043
and even here, I had to put
this little constraint here

00:06:17.043 --> 00:06:18.814
that x cannot be equal to negative 1.

00:06:18.814 --> 00:06:21.610
How do we think about this limit?

00:06:21.610 --> 00:06:23.043
Well, lucky for us,

00:06:23.043 --> 00:06:24.393
we know,

00:06:24.393 --> 00:06:26.262
lucky for us we know

00:06:26.262 --> 00:06:29.580
that if we just take
another function, f of x,

00:06:29.580 --> 00:06:33.747
if we say f of x is equal to
the square root of x plus 5

00:06:34.715 --> 00:06:35.548
plus 2,

00:06:36.443 --> 00:06:40.610
well then we know that f
of x is equal to g of x

00:06:42.003 --> 00:06:42.836
for all

00:06:44.363 --> 00:06:46.446
x not equal to negative 1

00:06:47.329 --> 00:06:49.649
because f of x does not
have that constraint.

00:06:49.649 --> 00:06:51.915
And we know if this is true of two,

00:06:51.915 --> 00:06:54.444
if this is true of two functions,

00:06:54.444 --> 00:06:57.037
then the limit as x approaches,

00:06:57.037 --> 00:06:59.088
the limit, let me write this down,

00:06:59.088 --> 00:07:00.622
since we know this,

00:07:00.622 --> 00:07:01.937
because of this,

00:07:01.937 --> 00:07:03.770
we know that the limit

00:07:04.659 --> 00:07:05.492
of f of x

00:07:06.330 --> 00:07:09.732
as x approaches negative
1 is going to be equal

00:07:09.732 --> 00:07:11.565
to the limit of g of x

00:07:12.748 --> 00:07:15.263
as x approaches negative 1,

00:07:15.263 --> 00:07:17.343
and this of course is what
we want to figure out,

00:07:17.343 --> 00:07:19.145
what was the beginning of the problem,

00:07:19.145 --> 00:07:21.269
but we can now use f of x here,

00:07:21.269 --> 00:07:23.095
because only at x equals negative 1

00:07:23.095 --> 00:07:25.223
that they are not the same,

00:07:25.223 --> 00:07:27.011
and if you were to graph g of x

00:07:27.011 --> 00:07:28.091
it just has a,

00:07:28.091 --> 00:07:29.885
it has a point discontinuity,

00:07:29.885 --> 00:07:31.141
or removable discont--

00:07:31.141 --> 00:07:32.786
or, I should say, yeah,

00:07:32.786 --> 00:07:35.780
a point discontinuity right over here

00:07:35.780 --> 00:07:37.947
at x equals negative 1,

00:07:38.848 --> 00:07:41.438
and so what is the limit?

00:07:41.438 --> 00:07:43.317
And we are in the home stretch now.

00:07:43.317 --> 00:07:45.102
What is the limit

00:07:45.102 --> 00:07:46.201
of f of x?

00:07:46.201 --> 00:07:49.941
Well, we could say the limit
of the square root of x plus 5

00:07:49.941 --> 00:07:50.774
plus 2

00:07:52.249 --> 00:07:54.749
as x approaches negative 1,

00:07:55.688 --> 00:07:58.676
well, this expression is continuous.

00:07:58.676 --> 00:08:01.658
Or, this function is continuous
at x equals negative 1

00:08:01.658 --> 00:08:03.938
so we can just evaluate
it at x equals negative 1,

00:08:03.938 --> 00:08:07.277
so this is going to be the
square root of negative 1

00:08:07.277 --> 00:08:08.444
plus 5 plus 2,

00:08:09.818 --> 00:08:12.059
so this is 4 square root,

00:08:12.059 --> 00:08:14.083
principle root of 4 is 2,

00:08:14.083 --> 00:08:16.000
2 plus 2 is equal to 4.

00:08:17.148 --> 00:08:20.504
So since the limit of f of
x as x approaches negative 1

00:08:20.504 --> 00:08:22.324
is 4, the limit of g of x

00:08:22.324 --> 00:08:26.120
as x approaches negative 1 is also 4,

00:08:26.120 --> 00:08:27.508
and if this little,

00:08:27.508 --> 00:08:28.428
this little,

00:08:28.428 --> 00:08:32.858
I guess you could say leap
that I just made over here

00:08:32.858 --> 00:08:33.828
doesn't make sense to you,

00:08:33.828 --> 00:08:34.728
think about it,

00:08:34.728 --> 00:08:36.257
think about it visually.

00:08:36.257 --> 00:08:38.487
Think about it visually.

00:08:38.487 --> 00:08:39.570
So if this is

00:08:40.879 --> 00:08:41.712
my y-axis

00:08:43.248 --> 00:08:45.081
and this is my x-axis,

00:08:46.869 --> 00:08:50.102
g of x looked something like this.

00:08:50.102 --> 00:08:50.935
G of x,

00:08:52.412 --> 00:08:54.663
g of x, let me draw it,

00:08:54.663 --> 00:08:56.913
g of x looked something,

00:08:58.580 --> 00:08:59.747
something like

00:09:00.606 --> 00:09:01.760
this,

00:09:01.760 --> 00:09:04.978
and it had a gap at negative 1,

00:09:04.978 --> 00:09:06.982
so it had a gap right over there,

00:09:06.982 --> 00:09:08.858
while f of x,

00:09:08.858 --> 00:09:10.778
f of x would have the same graph

00:09:10.778 --> 00:09:12.421
except it wouldn't have,

00:09:12.421 --> 00:09:14.067
it wouldn't have the gap,

00:09:14.067 --> 00:09:15.578
and so if you're trying to find the limit,

00:09:15.578 --> 00:09:17.010
it seems completely reasonable,

00:09:17.010 --> 00:09:19.225
well let's just use f
of x and evaluate what

00:09:19.225 --> 00:09:22.797
f of x would be to kind of fill that gap

00:09:22.797 --> 00:09:25.288
at x equals negative 1,

00:09:25.288 --> 00:09:27.711
so hopefully this graphical
version helps a little bit

00:09:27.711 --> 00:09:30.461
or if it confuses you, ignore it.


Undefined limits by direct substitution

WEBVTT
Kind: captions
Language: en

00:00:00.264 --> 00:00:01.650
- [Voiceover] Let's see if
we can figure out the limit

00:00:01.650 --> 00:00:05.521
of x over natural log of
x as x approaches one.

00:00:05.521 --> 00:00:07.443
And like always pause this
video and see if you can

00:00:07.443 --> 00:00:09.319
figure it out on your own.

00:00:09.319 --> 00:00:11.355
Well we know from out limit
properties this is going to

00:00:11.355 --> 00:00:14.001
be the same thing as the limit

00:00:14.001 --> 00:00:16.918
as x approaches one of x over

00:00:19.055 --> 00:00:20.721
over

00:00:20.721 --> 00:00:21.554
the limit,

00:00:23.370 --> 00:00:25.003
the limit

00:00:25.003 --> 00:00:27.645
as x approaches one

00:00:27.645 --> 00:00:29.567
of the natural log

00:00:29.567 --> 00:00:30.400
of x.

00:00:31.238 --> 00:00:33.962
Now this top limit, the
one I have in magenta,

00:00:33.962 --> 00:00:36.304
this is pretty straight forward,

00:00:36.304 --> 00:00:39.106
if we had the graph of y equals x

00:00:39.106 --> 00:00:40.857
that would be continuous everywhere

00:00:40.857 --> 00:00:42.768
it's defined for all real
numbers and it's continuous

00:00:42.768 --> 00:00:44.152
at all real numbers.

00:00:44.152 --> 00:00:47.491
So it's continuous to limit
as x approaches one of x.

00:00:47.491 --> 00:00:50.013
It's just gonna be this
evaluated x equals one.

00:00:50.013 --> 00:00:52.049
So this is just going to be one.

00:00:52.049 --> 00:00:54.447
We just put a one in for this x.

00:00:54.447 --> 00:00:57.697
For the numerator here we
just evaluate to a one.

00:00:57.697 --> 00:00:59.696
And then the denominator,

00:00:59.696 --> 00:01:03.888
natural log of x is not
defined for all x's,

00:01:03.888 --> 00:01:06.373
therefore it isn't continuous everywhere.

00:01:06.373 --> 00:01:10.000
But it is continuous at x equals one.

00:01:10.000 --> 00:01:13.094
And since it is continuous
at x equals one,

00:01:13.094 --> 00:01:15.334
then the limit here is just
gonna be the natural log

00:01:15.334 --> 00:01:17.245
evaluated at x equals one.

00:01:17.245 --> 00:01:19.770
So this is just going
to be the natural log

00:01:19.770 --> 00:01:22.171
the natural log of one.

00:01:22.171 --> 00:01:23.149
Which of course

00:01:23.149 --> 00:01:24.371
is zero.

00:01:24.371 --> 00:01:26.483
E to the zero power is one.

00:01:26.483 --> 00:01:29.659
So this is all going to be equal to

00:01:29.659 --> 00:01:31.981
this is going to be equal to

00:01:31.981 --> 00:01:33.328
we just evaluate it

00:01:33.328 --> 00:01:34.429
one

00:01:34.429 --> 00:01:35.772
over

00:01:35.772 --> 00:01:36.951
one

00:01:36.951 --> 00:01:37.931
over

00:01:37.931 --> 00:01:39.357
zero.

00:01:39.357 --> 00:01:41.518
And now we face a bit of a conundrum.

00:01:41.518 --> 00:01:44.075
One over zero is not defined.

00:01:44.075 --> 00:01:47.127
It is was zero over zero,
we wouldn't necessarily be

00:01:47.127 --> 00:01:49.163
done yet but it's indeterminate form

00:01:49.163 --> 00:01:51.968
as we will learn in the future
there are tools we can apply

00:01:51.968 --> 00:01:54.372
when we're trying to find
limits and we evaluate it

00:01:54.372 --> 00:01:56.361
like this and we get zero over zero.

00:01:56.361 --> 00:01:57.861
But one over zero.

00:01:58.846 --> 00:02:00.269
This is undefined

00:02:00.269 --> 00:02:01.901
which tells us that this limit

00:02:01.901 --> 00:02:03.151
does not exist.

00:02:04.302 --> 00:02:05.135
So does

00:02:06.788 --> 00:02:08.380
not

00:02:08.380 --> 00:02:09.803
exist.

00:02:09.803 --> 00:02:10.946
And

00:02:10.946 --> 00:02:11.946
we are done.


Limits by direct substitution

WEBVTT
Kind: captions
Language: en

00:00:00.703 --> 00:00:01.947
- [Voiceover] So let's see
if we can find the limit

00:00:01.947 --> 00:00:04.134
as X approaches negative
one of six X squared

00:00:04.134 --> 00:00:05.967
plus five X minus one.

00:00:07.359 --> 00:00:09.370
Now the first thing that
might jump out at you

00:00:09.370 --> 00:00:11.699
is this right over here,
this expression could be used

00:00:11.699 --> 00:00:14.965
to define the graph of a parabola.

00:00:14.965 --> 00:00:16.219
And when you think about
this, I'm not doing

00:00:16.219 --> 00:00:17.935
a rigorous proof here, parabola would look

00:00:17.935 --> 00:00:19.780
something like this and this would be an

00:00:19.780 --> 00:00:23.027
upward opening parabola,
look something like this.

00:00:23.027 --> 00:00:25.862
It is a, this graph,
visually, is continuous,

00:00:25.862 --> 00:00:28.006
you don't see any jumps or gaps in it.

00:00:28.006 --> 00:00:29.256
And in general,

00:00:31.702 --> 00:00:32.801
a quadratic like this

00:00:32.801 --> 00:00:35.777
is going to be defined for all values of X

00:00:35.777 --> 00:00:38.233
for all real numbers, and
it's going to be continuous,

00:00:38.233 --> 00:00:41.013
oh, for all real numbers.

00:00:41.013 --> 00:00:43.423
And so if something is
continuous for all real numbers,

00:00:43.423 --> 00:00:45.905
well then the limit as X
approaches some real number

00:00:45.905 --> 00:00:47.683
is going to be the same
thing, it's just a value

00:00:47.683 --> 00:00:50.535
within the expression at that real number.

00:00:50.535 --> 00:00:52.652
So what am I saying, I'm just
gonna say it another way,

00:00:52.652 --> 00:00:56.069
we know that some function is continuous,

00:00:57.151 --> 00:01:01.276
is continuous at some
X value, at X equals A

00:01:01.276 --> 00:01:05.193
if and only if, I'll
write that as iff, or iff,

00:01:06.048 --> 00:01:08.048
if and only if the limit

00:01:09.285 --> 00:01:11.535
as X approaches A of F of X

00:01:12.545 --> 00:01:13.962
is equal to F of,

00:01:14.967 --> 00:01:16.487
is equal to F of A.

00:01:16.487 --> 00:01:18.581
So I didn't do a rigorous proof here,

00:01:18.581 --> 00:01:21.431
but just, it's conceptually not a big jump

00:01:21.431 --> 00:01:23.632
to say, okay, well this
is, this is just a standard

00:01:23.632 --> 00:01:24.937
quadratic right over here,

00:01:24.937 --> 00:01:27.577
it's defined for all real numbers,

00:01:27.577 --> 00:01:30.528
and in fact it's continuous
for all real numbers.

00:01:30.528 --> 00:01:32.487
And so we know that this expression,

00:01:32.487 --> 00:01:34.451
it could define a continuous function,

00:01:34.451 --> 00:01:37.789
so that means that the
limit as X approaches A

00:01:37.789 --> 00:01:39.474
for this expression is just the same thing

00:01:39.474 --> 00:01:41.732
as evaluating this expression at A.

00:01:41.732 --> 00:01:44.175
And in this case, our A is negative one.

00:01:44.175 --> 00:01:46.262
So all I have to to is
evaluate this at negative one.

00:01:46.262 --> 00:01:49.583
This is going to be six
times negative one squared

00:01:49.583 --> 00:01:52.833
plus five times negative one minus one.

00:01:53.839 --> 00:01:57.399
So that's just one, this is negative five,

00:01:57.399 --> 00:02:00.048
so it's six minus five minus one,

00:02:00.048 --> 00:02:02.266
which is equal to

00:02:02.266 --> 00:02:03.212
zero.

00:02:03.212 --> 00:02:04.545
And we are done.


Functions continuous at specific x-values

WEBVTT
Kind: captions
Language: en

00:00:00.250 --> 00:00:01.712
- [Voiceover] Which of
the following functions

00:00:01.712 --> 00:00:04.712
are continuous at x equals three?

00:00:05.598 --> 00:00:07.332
Well, as we said in the previous video,

00:00:07.332 --> 00:00:08.435
in the previous example,

00:00:08.435 --> 00:00:10.277
in order to be continuous at a point,

00:00:10.277 --> 00:00:12.875
you at least have to be
defined at that point.

00:00:12.875 --> 00:00:14.523
We saw our definition of continuity,

00:00:14.523 --> 00:00:17.215
f is continuous at a, if and only if,

00:00:17.215 --> 00:00:21.638
the limit of f as x approaches
a is equal to f of a.

00:00:21.638 --> 00:00:23.390
So, over here, in this case,

00:00:23.390 --> 00:00:25.300
we could say that a function is continuous

00:00:25.300 --> 00:00:29.467
at x equals three, so f is
continuous at x equals three,

00:00:33.199 --> 00:00:37.366
if and only if the limit as
x approaches three of f of x,

00:00:40.991 --> 00:00:42.908
is equal to f of three.

00:00:43.979 --> 00:00:47.261
Now let's look at this first
function right over here.

00:00:47.261 --> 00:00:49.702
Natural log of x minus three.

00:00:49.702 --> 00:00:52.271
Well, try to evaluate it,
and it's not an f now,

00:00:52.271 --> 00:00:54.830
it's g, try to evaluate g of three.

00:00:54.830 --> 00:00:57.155
G of three, let me write it here,

00:00:57.155 --> 00:01:01.159
g of three is equal to
the natural log of zero.

00:01:01.159 --> 00:01:02.125
Three minus three.

00:01:02.125 --> 00:01:03.472
This is not defined.

00:01:03.472 --> 00:01:06.856
You can't raise e to any
power to get to zero.

00:01:06.856 --> 00:01:09.392
You can try to go to, you
could say, negative infinity,

00:01:09.392 --> 00:01:11.489
but that's not, this is not defined.

00:01:11.489 --> 00:01:14.801
And so, if this isn't even
defined at x equals three,

00:01:14.801 --> 00:01:16.607
there's no way that it's
going to be continuous

00:01:16.607 --> 00:01:20.774
at x equals three, so we
could rule this one out.

00:01:21.686 --> 00:01:24.493
Now f of x is equal to
e to the x minus three.

00:01:24.493 --> 00:01:27.053
Well this is just a shifted
over version of e to the x.

00:01:27.053 --> 00:01:30.181
This is defined for all real numbers,

00:01:30.181 --> 00:01:32.618
and as we saw in the previous example,

00:01:32.618 --> 00:01:35.550
it's reasonable to say it's
continuous for all real numbers,

00:01:35.550 --> 00:01:38.071
and you could even do
this little test here.

00:01:38.071 --> 00:01:41.990
The limit of e to the x minus
three as x approaches three,

00:01:41.990 --> 00:01:43.756
well that is going to
be, that is going to be

00:01:43.756 --> 00:01:46.994
e to the three minus three,
or e to the zero, or one.

00:01:46.994 --> 00:01:50.661
And so f is the only
one that is continuous.

00:01:52.019 --> 00:01:53.610
And once again, it's good to think about

00:01:53.610 --> 00:01:56.961
what's going on here
visually, if you like.

00:01:56.961 --> 00:01:58.752
Both of these are, you
could think of them,

00:01:58.752 --> 00:02:00.796
this is a shifted over version of ln of x,

00:02:00.796 --> 00:02:04.211
this is a shifted over
version of e to the x,

00:02:04.211 --> 00:02:08.132
and so if we like, we could
draw ourselves some axes,

00:02:08.132 --> 00:02:11.549
so that's our y-axis, this is our x-axis,

00:02:16.114 --> 00:02:17.441
and actually, let me
draw some points here.

00:02:17.441 --> 00:02:21.358
So that's one, that is
one, that is two, three,

00:02:24.571 --> 00:02:27.071
two, and three, and let's see,

00:02:29.765 --> 00:02:32.080
I said these are shifted over versions,

00:02:32.080 --> 00:02:35.276
so actually, this is maybe
not the best way to draw it,

00:02:35.276 --> 00:02:38.859
so let me draw it, this
is one, two, three,

00:02:40.800 --> 00:02:42.467
four, five, and six.

00:02:44.636 --> 00:02:46.723
And on this axis, I won't
make 'em on the same scale,

00:02:46.723 --> 00:02:49.556
let's say this is one, two, three.

00:02:50.394 --> 00:02:52.073
I'm gonna draw one, two, three,

00:02:52.073 --> 00:02:55.398
I'm gonna draw a dotted
line right over here.

00:02:55.398 --> 00:02:58.745
So g of x, ln of x minus three

00:02:58.745 --> 00:03:00.282
is gonna look something like this.

00:03:00.282 --> 00:03:01.729
If you put three in it, it's not defined,

00:03:01.729 --> 00:03:04.099
if you put four in it, ln of four,

00:03:04.099 --> 00:03:07.177
well, that's gonna, sorry,
ln of four minus one,

00:03:07.177 --> 00:03:10.677
so that's gonna be ln of four minus three,

00:03:12.015 --> 00:03:14.890
is actually let me just draw a table here,

00:03:14.890 --> 00:03:16.845
I know I'm confusing you.

00:03:16.845 --> 00:03:19.762
So, if I say x and I say g of x,

00:03:20.859 --> 00:03:23.359
so at three, you're undefined.

00:03:24.982 --> 00:03:28.149
At four, this is ln of one, ln of one,

00:03:30.584 --> 00:03:33.925
which is equal to zero,
so it's right over there.

00:03:33.925 --> 00:03:38.092
So g of x is gonna look something
like, something like that.

00:03:40.213 --> 00:03:41.937
And so you can see at three,

00:03:41.937 --> 00:03:44.674
you have this discontinuity there,

00:03:44.674 --> 00:03:46.964
it's not even defined
to the left of three.

00:03:46.964 --> 00:03:50.318
Now f of x is a little
bit more straightforward.

00:03:50.318 --> 00:03:54.151
If you have, so e to the
three is going to be,

00:03:56.709 --> 00:04:01.181
sorry, f of three is going to
be e to the three minus three,

00:04:01.181 --> 00:04:03.751
or e to the zero, so it's going to be one,

00:04:03.751 --> 00:04:06.525
so it's gonna look something like this,

00:04:06.525 --> 00:04:09.138
it's gonna look something,

00:04:09.138 --> 00:04:11.305
something like, like that.

00:04:12.755 --> 00:04:14.498
There's no jumps, there's no gaps,

00:04:14.498 --> 00:04:17.148
it is going to be continuous, and frankly,

00:04:17.148 --> 00:04:18.743
all real numbers so for sure it's going

00:04:18.743 --> 00:04:20.872
to be continuous at three.


Functions continuous on all real numbers

WEBVTT
Kind: captions
Language: en

00:00:00.459 --> 00:00:02.210
- [Voiceover] Which of
the following functions

00:00:02.210 --> 00:00:05.210
are continuous for all real numbers?

00:00:07.401 --> 00:00:08.834
So let's just remind ourselves

00:00:08.834 --> 00:00:10.518
what it means to be continuous.

00:00:10.518 --> 00:00:13.512
What a continuous function looks like.

00:00:13.512 --> 00:00:15.176
So, a continuous function,

00:00:15.176 --> 00:00:17.425
let's see, that's my y-axis,

00:00:17.425 --> 00:00:19.678
that is my x-axis.

00:00:19.678 --> 00:00:22.423
A function is going to be
continuous over some interval.

00:00:22.423 --> 00:00:23.256
If it just has,

00:00:23.256 --> 00:00:26.831
doesn't have any jumps or discontinuities

00:00:26.831 --> 00:00:29.300
over that, or gaps over that interval,

00:00:29.300 --> 00:00:31.024
so if it's connected

00:00:31.024 --> 00:00:33.721
and it for sure has to be
defined over that interval

00:00:33.721 --> 00:00:35.259
without any gaps,

00:00:35.259 --> 00:00:36.092
so for example,

00:00:36.092 --> 00:00:40.259
a continuous function could
look something like this.

00:00:41.727 --> 00:00:42.630
This function,

00:00:42.630 --> 00:00:44.810
let me make that line
a little bit thicker,

00:00:44.810 --> 00:00:47.888
so this function right
over here is continuous.

00:00:47.888 --> 00:00:50.226
It is connected over this interval,

00:00:50.226 --> 00:00:51.526
the interval that we can see.

00:00:51.526 --> 00:00:53.899
Now, examples of discontinuous functions

00:00:53.899 --> 00:00:56.425
over an interval, or
non-continuous functions,

00:00:56.425 --> 00:00:58.223
well, they would have gaps of some kind.

00:00:58.223 --> 00:01:01.712
They could have some type of
an asymptotic discontinuity

00:01:01.712 --> 00:01:03.749
so something like that,

00:01:03.749 --> 00:01:06.495
that makes it discontinuous.

00:01:06.495 --> 00:01:08.122
They could have

00:01:08.122 --> 00:01:09.516
a jump of discontinuity,

00:01:09.516 --> 00:01:11.183
something like that.

00:01:12.416 --> 00:01:13.880
They could just have a gap

00:01:13.880 --> 00:01:16.047
where they're not defined,

00:01:17.044 --> 00:01:19.989
so they could have a gap
where they're not defined,

00:01:19.989 --> 00:01:22.845
or maybe they actually are defined there,

00:01:22.845 --> 00:01:25.573
but it's removable discontinuity,

00:01:25.573 --> 00:01:28.761
so all of these are examples
of discontinuous functions.

00:01:28.761 --> 00:01:31.492
Now, if you want the more
mathy understanding of that

00:01:31.492 --> 00:01:32.988
and we've looked at this before,

00:01:32.988 --> 00:01:36.238
we say that a function f is continuous,

00:01:37.122 --> 00:01:38.622
continuous at some

00:01:39.873 --> 00:01:41.556
value, x equals a,

00:01:41.556 --> 00:01:42.940
if and only if,

00:01:42.940 --> 00:01:44.626
draw my little two-way arrows here,

00:01:44.626 --> 00:01:47.640
say if and only if the limit

00:01:47.640 --> 00:01:50.390
of f of x as x approaches a

00:01:51.380 --> 00:01:55.573
is equal to the value
of the function at a,

00:01:55.573 --> 00:01:57.661
so once again, in order
to be continuous there,

00:01:57.661 --> 00:02:00.590
you at least have to be defined there.

00:02:00.590 --> 00:02:02.439
Now, when you look at these,
the one thing that jumps out

00:02:02.439 --> 00:02:04.182
at me, in order to be
continuous for all real numbers,

00:02:04.182 --> 00:02:05.929
you have to be defined
for all real numbers

00:02:05.929 --> 00:02:08.772
and g of x is not defined
for all real numbers.

00:02:08.772 --> 00:02:11.857
It's not defined for negative values of x,

00:02:11.857 --> 00:02:13.190
and so, we would

00:02:14.210 --> 00:02:16.282
rule this one out,

00:02:16.282 --> 00:02:19.317
so let's think about f
of x equals e to the x.

00:02:19.317 --> 00:02:22.234
It is defined for all real numbers,

00:02:23.439 --> 00:02:24.604
and as we'll see,

00:02:24.604 --> 00:02:27.349
most of the common functions
that you've learned in math,

00:02:27.349 --> 00:02:30.020
they don't have these
strange jumps or gaps

00:02:30.020 --> 00:02:31.007
or discontinuities.

00:02:31.007 --> 00:02:32.415
Some of them do,

00:02:32.415 --> 00:02:34.104
functions like 1 over x

00:02:34.104 --> 00:02:36.020
and things like that,

00:02:36.020 --> 00:02:39.210
but things like e to the x,
it doesn't have any of those.

00:02:39.210 --> 00:02:40.568
We could graph e to the x.

00:02:40.568 --> 00:02:42.850
E to the x looks something like,

00:02:42.850 --> 00:02:45.228
e to the x looks something like this,

00:02:45.228 --> 00:02:47.009
it's defined for all real numbers,

00:02:47.009 --> 00:02:50.182
there's no jumps or gaps of any kind

00:02:50.182 --> 00:02:51.384
and so,

00:02:51.384 --> 00:02:54.276
this f of x is continuous

00:02:54.276 --> 00:02:55.732
for all real numbers

00:02:55.732 --> 00:02:57.307
and f only.

00:02:57.307 --> 00:02:59.226
Now, I didn't do a very rigorous proof.

00:02:59.226 --> 00:03:00.816
You could if you like,

00:03:00.816 --> 00:03:02.809
but for the sake of this exercise,

00:03:02.809 --> 00:03:04.499
it's really more of getting
this intuitive sense

00:03:04.499 --> 00:03:07.499
of like, look, e to the x is
defined for all real numbers

00:03:07.499 --> 00:03:09.139
and so,

00:03:09.139 --> 00:03:11.202
and there's no jumps or gaps here

00:03:11.202 --> 00:03:12.648
so it's reasonable to
say that it's continuous

00:03:12.648 --> 00:03:14.548
but you could do a more
rigorous proof if you like

00:03:14.548 --> 00:03:15.381
as well.


Limits of composite functions

WEBVTT
Kind: captions
Language: en

00:00:00.440 --> 00:00:02.340
- [Voiceover] Let's now
take some limits involving

00:00:02.340 --> 00:00:04.484
composite functions.

00:00:04.484 --> 00:00:07.277
So over here we have
the limit of g of h of x

00:00:07.277 --> 00:00:10.003
as x approaches three, and like always,

00:00:10.003 --> 00:00:12.027
I encourage you to pause
the video and see if you can

00:00:12.027 --> 00:00:14.065
figure this out on your own.

00:00:14.065 --> 00:00:16.088
Well, we can leverage
our limit properties,

00:00:16.088 --> 00:00:17.965
we know that this is
going to be the same thing

00:00:17.965 --> 00:00:20.767
as the limit, actually
let me write it this way,

00:00:20.767 --> 00:00:24.934
this is going to be the
same thing as g of the limit

00:00:26.253 --> 00:00:29.170
as x approaches three of h of x,

00:00:30.019 --> 00:00:34.070
or I could say the limit of
h of x as x approaches three.

00:00:34.070 --> 00:00:35.920
And so we just need to figure out

00:00:35.920 --> 00:00:39.885
what the limit of h of x
is as x approaches three.

00:00:39.885 --> 00:00:42.331
So, let's look at h of x right over here

00:00:42.331 --> 00:00:45.088
and as x approaches three, so we see that

00:00:45.088 --> 00:00:48.954
h of three is undefined
but we can think about

00:00:48.954 --> 00:00:52.831
what the limit of h of x
is as x approaches three.

00:00:52.831 --> 00:00:56.175
As x approaches three from the left,

00:00:56.175 --> 00:00:58.388
as x approaches three from the left,

00:00:58.388 --> 00:01:01.785
we see that the function
just stays at a constant two,

00:01:01.785 --> 00:01:05.035
so at h of 2.5 is two, h of 2.9 is two,

00:01:07.899 --> 00:01:09.899
h of 2.999999999 is two.

00:01:11.152 --> 00:01:12.838
So it looks like when we
approach from the left,

00:01:12.838 --> 00:01:15.159
the limit is two and we
approach from the right,

00:01:15.159 --> 00:01:19.023
we get the same thing, h of 3.01 is two,

00:01:19.023 --> 00:01:22.440
h of 3.001 is two, h of 3.0000001 is two.

00:01:25.365 --> 00:01:28.891
So this limit right over here is two.

00:01:28.891 --> 00:01:32.058
So this is all simplified to g of two.

00:01:33.196 --> 00:01:34.508
Now what is g of two?

00:01:34.508 --> 00:01:36.435
Well, let's see, this function here,

00:01:36.435 --> 00:01:39.352
when x is two, g of two is zero,

00:01:40.973 --> 00:01:44.635
so this right over there
is going to be zero,

00:01:44.635 --> 00:01:48.234
and we're done, let's
do a few more of these.

00:01:48.234 --> 00:01:52.032
Alright, so we wanna find
the limit as x approaches

00:01:52.032 --> 00:01:54.536
negative one of h of g of x.

00:01:54.536 --> 00:01:56.517
Well, just like we just
did, this is going to be

00:01:56.517 --> 00:02:00.434
the same thing, this is
equal to h of the limit

00:02:04.726 --> 00:02:08.226
as x approaches negative one of g of x,

00:02:11.460 --> 00:02:13.762
so let's try to figure
out the limit of g of x

00:02:13.762 --> 00:02:15.817
as x approaches negative one.

00:02:15.817 --> 00:02:18.764
So, this is the graph y equals g of x

00:02:18.764 --> 00:02:22.386
and we see at negative
one right over here,

00:02:22.386 --> 00:02:26.219
we have this discontinuity
and as we approach,

00:02:27.113 --> 00:02:30.591
as we approach x equals
negative one from the left,

00:02:30.591 --> 00:02:34.441
it looks like we go unbounded
in the negative direction,

00:02:34.441 --> 00:02:37.247
so you could say we're
approaching negative infinity,

00:02:37.247 --> 00:02:40.729
and as we go from the right,
as we go from the right,

00:02:40.729 --> 00:02:43.725
looks like we are, as we
get closer to x equals

00:02:43.725 --> 00:02:45.830
negative one on the right-hand side,

00:02:45.830 --> 00:02:48.094
looks like we're approaching infinity,

00:02:48.094 --> 00:02:51.149
so even if they were both
approaching the same direction

00:02:51.149 --> 00:02:53.553
of infinity, we would say
that the limit's not defined

00:02:53.553 --> 00:02:56.300
or at least that's the
technical idea here.

00:02:56.300 --> 00:02:59.063
But this is going, one's going
towards positive infinity,

00:02:59.063 --> 00:03:00.724
and the other is going
to negative infinity

00:03:00.724 --> 00:03:03.265
so this limit right here is undefined.

00:03:03.265 --> 00:03:07.432
So, it doesn't exist, or I
should say, does not exist.

00:03:09.232 --> 00:03:12.616
So, if the limit as x approaches
negative one of g of x

00:03:12.616 --> 00:03:14.505
does not exist, well,
there's no way we that

00:03:14.505 --> 00:03:16.107
we can evaluate this expression.

00:03:16.107 --> 00:03:18.230
We can't find h of does not exist,

00:03:18.230 --> 00:03:21.027
so this entire limit does not,

00:03:21.027 --> 00:03:23.651
this entire limit does not exist.

00:03:23.651 --> 00:03:26.023
Let's do one more of these.

00:03:26.023 --> 00:03:30.190
Alright, so we have once
again limit of h of f of x

00:03:32.510 --> 00:03:35.964
as x approaches negative
three, this is the same thing,

00:03:35.964 --> 00:03:38.547
this is equal to h of the limit

00:03:40.324 --> 00:03:43.991
as x approaches negative three of f of x.

00:03:47.236 --> 00:03:51.181
So, let's look at f of x, this
is the graph y equals f of x,

00:03:51.181 --> 00:03:54.009
and the limit as x
approaches negative three,

00:03:54.009 --> 00:03:58.449
well, as we approach negative
three from the left-hand side

00:03:58.449 --> 00:04:01.336
as we get closer and
closer to negative three,

00:04:01.336 --> 00:04:04.182
it looks like we are
approaching the value of one,

00:04:04.182 --> 00:04:06.337
and as we approach from
the right-hand side,

00:04:06.337 --> 00:04:08.601
it looks like we're
approaching the value of one.

00:04:08.601 --> 00:04:10.265
If I were to take from the left-hand side,

00:04:10.265 --> 00:04:13.653
if I were to take negative
3.1, negative 3.01,

00:04:13.653 --> 00:04:17.069
negative 3.001, I'm gonna
get closer and closer to,

00:04:17.069 --> 00:04:19.031
if I evaluate the function
there, so I should take

00:04:19.031 --> 00:04:22.825
f of negative 3.1, f of negative 3.01,

00:04:22.825 --> 00:04:25.699
f of negative 3.0001, we're getting

00:04:25.699 --> 00:04:27.581
closer and closer to one, and same thing

00:04:27.581 --> 00:04:31.007
on the right-hand side, so
this thing looks like it's one.

00:04:31.007 --> 00:04:34.277
So, now we just have to
evaluate, and I'll rewrite it,

00:04:34.277 --> 00:04:37.444
so this is the same thing as h of one,

00:04:38.903 --> 00:04:41.216
h of one, so you just
have to evaluate this.

00:04:41.216 --> 00:04:44.752
Then when we look at
this graph here at one,

00:04:44.752 --> 00:04:47.509
this function does not look defined,

00:04:47.509 --> 00:04:51.676
so h of one is actually
undefined, undefined right here,

00:04:55.125 --> 00:04:58.931
so also in this case, this
limit would not exist.

00:04:58.931 --> 00:05:00.948
Once again, the limit
part was actually at least

00:05:00.948 --> 00:05:03.747
the limit of f of x was
fairly straight forward,

00:05:03.747 --> 00:05:05.915
but then when we tried to take that output

00:05:05.915 --> 00:05:07.633
and put it as the input into h of x,

00:05:07.633 --> 00:05:10.800
well, h of x, h wasn't defined there.


Worked example: Continuity at a point

WEBVTT
Kind: captions
Language: en

00:00:00.403 --> 00:00:04.283
- [Voiceover] We have the
graph of y is equal to g of x

00:00:04.283 --> 00:00:07.393
right over here, and what
I wanna do is I wanna check

00:00:07.393 --> 00:00:10.769
which of these statements
are actually true

00:00:10.769 --> 00:00:12.158
and then check them off.

00:00:12.158 --> 00:00:14.592
And like always, I encourage
you to pause the video

00:00:14.592 --> 00:00:17.438
and see if you can work
through this on your own.

00:00:17.438 --> 00:00:20.169
So let's look at this first statement.

00:00:20.169 --> 00:00:23.045
So this first statement says
both the limit of g of x

00:00:23.045 --> 00:00:26.437
as x approaches six
from the right-hand side

00:00:26.437 --> 00:00:29.946
and the limit as x approaches
six from the left-hand side

00:00:29.946 --> 00:00:31.529
of g of x exist.

00:00:32.364 --> 00:00:34.729
All right, so let's first
think about the limit

00:00:34.729 --> 00:00:39.339
of g of x as x approaches
six from the right-hand side,

00:00:39.339 --> 00:00:42.701
as we approach six from
values greater than six.

00:00:42.701 --> 00:00:45.093
So if we look over
here, we could say okay,

00:00:45.093 --> 00:00:48.374
when x is equal to nine, and
g of nine is right over there,

00:00:48.374 --> 00:00:50.592
g of eight is right over here,

00:00:50.592 --> 00:00:53.464
g of seven is right over here,

00:00:53.464 --> 00:00:56.240
looks like it's between negative
three and negative four,

00:00:56.240 --> 00:00:59.407
g of 6.5 looks like it's a little bit,

00:01:01.326 --> 00:01:04.432
it's still between negative
three and negative four

00:01:04.432 --> 00:01:06.126
but it's closer to negative three.

00:01:06.126 --> 00:01:09.626
G of 6.1 is even closer to negative three.

00:01:11.088 --> 00:01:14.671
G of 6.01 is even closer
to negative three,

00:01:17.174 --> 00:01:18.885
so it looks like the limit

00:01:18.885 --> 00:01:21.332
from the right-hand side does exist.

00:01:21.332 --> 00:01:23.817
So it looks like this one exists.

00:01:23.817 --> 00:01:26.035
Now let's see, and I'm just
looking at it graphically.

00:01:26.035 --> 00:01:27.741
That's all they can expect you to do

00:01:27.741 --> 00:01:29.080
in an exercise like this.

00:01:29.080 --> 00:01:31.441
Now let's think about the
limit as x approaches six

00:01:31.441 --> 00:01:34.192
from the left-hand side.

00:01:34.192 --> 00:01:37.037
So I could start anywhere,
but let's say when x is equal

00:01:37.037 --> 00:01:41.524
to three, g of three is
a little more than one.

00:01:41.524 --> 00:01:45.703
G of four looks like there's
a little bit less than two.

00:01:45.703 --> 00:01:48.207
G of five looks like it's close to three.

00:01:48.207 --> 00:01:52.040
G of 5.5 looks like it's
between five and six.

00:01:53.692 --> 00:01:57.275
G of 5.75 looks like
it's approaching nine.

00:01:59.471 --> 00:02:00.888
And as we get closer and closer,

00:02:00.888 --> 00:02:04.567
as x gets closer and
closer to six from below,

00:02:04.567 --> 00:02:06.649
from values to the left of six,

00:02:06.649 --> 00:02:10.220
it looks like we're unbounded,
we're approaching infinity.

00:02:10.220 --> 00:02:13.432
And so technically, we would
say this limit does not exist.

00:02:13.432 --> 00:02:15.204
So this one does not exist.

00:02:15.204 --> 00:02:16.918
So I won't check this one off.

00:02:16.918 --> 00:02:18.827
Some people say the limit
is approaching infinity,

00:02:18.827 --> 00:02:20.827
but that technically is,

00:02:22.088 --> 00:02:25.638
infinity is not a value that you can say

00:02:25.638 --> 00:02:28.481
it is approaching in the
classical formal definition

00:02:28.481 --> 00:02:29.482
of a limit.

00:02:29.482 --> 00:02:31.700
So for these purposes, we would just say

00:02:31.700 --> 00:02:33.162
this does not exist.

00:02:33.162 --> 00:02:36.564
Now let's see, they say the
limit as x approaches six

00:02:36.564 --> 00:02:38.085
of g of x exists.

00:02:38.085 --> 00:02:40.387
Well, the only way that the limit exists

00:02:40.387 --> 00:02:44.113
is if both the left, if
both the left, the left,

00:02:44.113 --> 00:02:48.510
and the right limits exist and
they approach the same thing.

00:02:48.510 --> 00:02:53.467
We'll, our limit as x approaches
six from the negative side

00:02:53.467 --> 00:02:55.533
or from the left-hand
side, I guess I could say,

00:02:55.533 --> 00:02:56.781
does not even exist.

00:02:56.781 --> 00:02:58.256
So this cannot be true.

00:02:58.256 --> 00:02:59.773
So that's not gonna be true.

00:02:59.773 --> 00:03:01.247
The first one's not gonna be true.

00:03:01.247 --> 00:03:04.184
G is defined as x equals six.

00:03:04.184 --> 00:03:07.775
So at x equals six, it doesn't
look like g is defined.

00:03:07.775 --> 00:03:09.352
Looking at this graph, I can't tell you

00:03:09.352 --> 00:03:10.544
what g of six should be.

00:03:10.544 --> 00:03:12.775
We have an open circle
over here, so g of six

00:03:12.775 --> 00:03:14.763
does not equal to negative three,

00:03:14.763 --> 00:03:16.242
and this goes up to infinity,

00:03:16.242 --> 00:03:18.483
and we have a vertical
asymptote actually drawn

00:03:18.483 --> 00:03:20.436
right over here at x equals six.

00:03:20.436 --> 00:03:22.691
So g is not defined at x equals six.

00:03:22.691 --> 00:03:24.329
So I'll rule that one out.

00:03:24.329 --> 00:03:27.207
G is continuous at x equals six.

00:03:27.207 --> 00:03:29.650
Well, you can see that
it goes up to infinity

00:03:29.650 --> 00:03:32.587
then it jumps down, back
down here, then it continues.

00:03:32.587 --> 00:03:35.296
So when you just think about
it in commonsense language,

00:03:35.296 --> 00:03:37.498
it looks very discontinuous.

00:03:37.498 --> 00:03:39.186
And if you wanna think
about it more formally,

00:03:39.186 --> 00:03:41.576
in order for something to be continuous,

00:03:41.576 --> 00:03:44.934
the limit needs to exist at that value.

00:03:44.934 --> 00:03:48.109
The function needs to be
defined at that value,

00:03:48.109 --> 00:03:50.513
and the value of the
function needs to be equal

00:03:50.513 --> 00:03:52.079
to the value of the limit

00:03:52.079 --> 00:03:56.165
and neither of these, the
first two conditions are true

00:03:56.165 --> 00:03:58.220
and so these can't even equal
each other because neither

00:03:58.220 --> 00:03:59.967
of these exist.

00:03:59.967 --> 00:04:02.480
So this is not continuous at x equals six

00:04:02.480 --> 00:04:07.020
and so the only think I can
check here is none of the above.

00:04:07.020 --> 00:04:09.975
Let's do another one of these.

00:04:09.975 --> 00:04:13.158
So the first statement,
both the right-hand

00:04:13.158 --> 00:04:16.630
and the left-hand limit
exists as x approaches three.

00:04:16.630 --> 00:04:17.850
So let's think about it.

00:04:17.850 --> 00:04:20.998
So x equals three is
where we have this little

00:04:20.998 --> 00:04:24.144
discontinuity here,
this jump discontinuity.

00:04:24.144 --> 00:04:26.918
So let's approach, let's
go from the positive,

00:04:26.918 --> 00:04:28.469
from values larger than three.

00:04:28.469 --> 00:04:33.056
So when x is equal to five,
g of five is a little bit

00:04:33.056 --> 00:04:34.950
more negative than negative three.

00:04:34.950 --> 00:04:38.337
G of four is between negative
two and negative three.

00:04:38.337 --> 00:04:42.420
G of 3.5 is getting a bit
closer to negative two.

00:04:43.489 --> 00:04:46.406
G of 3.1, it's getting even closer,

00:04:48.647 --> 00:04:50.220
closer to negative two.

00:04:50.220 --> 00:04:53.637
G of 3.01 is even closer to negative two.

00:04:54.830 --> 00:04:57.259
So it looks like this
limit right over here,

00:04:57.259 --> 00:04:58.864
well, I'm circling the wrong one,

00:04:58.864 --> 00:05:00.866
it looks like this limit exists.

00:05:00.866 --> 00:05:04.914
In fact, it looks like it
is approaching negative two.

00:05:04.914 --> 00:05:07.894
So this right over here
is equal to negative two.

00:05:07.894 --> 00:05:10.302
The limit of g of x as x approaches three

00:05:10.302 --> 00:05:12.887
from the right-hand side,

00:05:12.887 --> 00:05:16.252
and I'll just think about
it from the left-hand side.

00:05:16.252 --> 00:05:17.919
So I can start here.

00:05:19.283 --> 00:05:22.901
G of one, looks like it's a little bit

00:05:22.901 --> 00:05:25.605
greater than negative one.

00:05:25.605 --> 00:05:28.022
G of two, it's less than one.

00:05:29.760 --> 00:05:32.839
G of 2.5 is between one and two.

00:05:32.839 --> 00:05:37.378
G of 2.9 looks like it's a
little bit less than two.

00:05:37.378 --> 00:05:41.408
G of 2.99 is getting even closer to two.

00:05:41.408 --> 00:05:44.477
G of 2.99999 would be even closer to two

00:05:44.477 --> 00:05:46.352
so it looks like this
thing right over here

00:05:46.352 --> 00:05:47.538
is approaching two.

00:05:47.538 --> 00:05:50.074
So both of these limits,
the limit from the right

00:05:50.074 --> 00:05:52.907
and the limit from the left exist.

00:05:53.882 --> 00:05:58.160
The limit of g of x as x
approaches three exists.

00:05:58.160 --> 00:06:00.250
So these are the one-sided limit.

00:06:00.250 --> 00:06:02.120
This is the actual limit.

00:06:02.120 --> 00:06:05.126
Now in order for this
to exist, both the right

00:06:05.126 --> 00:06:07.937
and left-handed limits need to exist

00:06:07.937 --> 00:06:10.090
and they need to approach the same value.

00:06:10.090 --> 00:06:12.094
Well, this first statement,

00:06:12.094 --> 00:06:15.126
we saw that both of these exist
but they aren't approaching

00:06:15.126 --> 00:06:16.129
the same value.

00:06:16.129 --> 00:06:19.118
From the left, we are, or sorry,

00:06:19.118 --> 00:06:21.370
from the right, we are approaching,

00:06:21.370 --> 00:06:23.401
we are approaching negative two.

00:06:23.401 --> 00:06:26.178
And from the left, we are approaching two.

00:06:26.178 --> 00:06:28.748
So this limit does not exist.

00:06:28.748 --> 00:06:32.451
So I will not check that out
or I will not check that box.

00:06:32.451 --> 00:06:35.653
G is defined at x equals three.

00:06:35.653 --> 00:06:38.500
Well, when x equals
three, we see a solid dot

00:06:38.500 --> 00:06:39.744
right over there.

00:06:39.744 --> 00:06:41.797
And so it is indeed defined.

00:06:41.797 --> 00:06:44.506
It is indeed defined there.

00:06:44.506 --> 00:06:47.040
G is continuous at x equals three.

00:06:47.040 --> 00:06:50.468
Well, in order for g to be
continuous at x equals three,

00:06:50.468 --> 00:06:53.158
the limit must exist there.

00:06:53.158 --> 00:06:54.994
It must be defined there,

00:06:54.994 --> 00:06:57.210
and the value of the function there

00:06:57.210 --> 00:06:59.595
needs to be equal to
the value of the limit.

00:06:59.595 --> 00:07:02.192
Well, the function is defined there,

00:07:02.192 --> 00:07:03.800
but the limit doesn't exist there

00:07:03.800 --> 00:07:06.397
so it cannot be continuous.

00:07:06.397 --> 00:07:07.766
It cannot be continuous there.

00:07:07.766 --> 00:07:09.308
So I would cross that out.

00:07:09.308 --> 00:07:13.272
And I can't click, I wouldn't
click none of the above

00:07:13.272 --> 00:07:15.231
because I've already checked something,

00:07:15.231 --> 00:07:17.854
or I've actually checked
two things already.


One-sided limits from graphs: asymptote

WEBVTT
Kind: captions
Language: en

00:00:00.364 --> 00:00:02.130
- [Voiceover] Over here
we have the graph of

00:00:02.130 --> 00:00:04.451
y is equal to g of x.

00:00:04.451 --> 00:00:05.997
What I wanna do is I wanna figure out

00:00:05.997 --> 00:00:07.580
the limit of g of x

00:00:09.398 --> 00:00:11.981
as x approaches positive six

00:00:14.457 --> 00:00:18.040
from values that are
less than positive six

00:00:18.960 --> 00:00:21.498
or you could say from the
left, from the, you could say

00:00:21.498 --> 00:00:23.422
the negative direction.

00:00:23.422 --> 00:00:25.121
So what is this going to be equal to?

00:00:25.121 --> 00:00:26.711
And if you have a sense
of it, pause the video

00:00:26.711 --> 00:00:28.852
and give a go at it.

00:00:28.852 --> 00:00:30.651
Well, to think about this,

00:00:30.651 --> 00:00:33.726
let's just take different
x-values that approach six

00:00:33.726 --> 00:00:36.357
from the left and look at what the values

00:00:36.357 --> 00:00:37.727
of the function are.

00:00:37.727 --> 00:00:41.187
So g of two, looks like it's
a little bit more than one.

00:00:41.187 --> 00:00:43.939
G of three, it's a little
bit more than that.

00:00:43.939 --> 00:00:46.809
G of four, looks like
it's a little under two.

00:00:46.809 --> 00:00:49.576
G of five, it looks
like it's around three.

00:00:49.576 --> 00:00:52.743
G of 5.5, looks like it's around five.

00:00:53.924 --> 00:00:55.674
G of, let's say 5.75,

00:00:57.685 --> 00:00:59.852
looks like it's like nine.

00:01:00.740 --> 00:01:03.758
And so, as x gets closer and closer to six

00:01:03.758 --> 00:01:07.083
from the left, it looks like
the value of our function

00:01:07.083 --> 00:01:11.451
becomes unbounded, it's just
getting infinitely large.

00:01:11.451 --> 00:01:13.995
And so in some context,
you might see someone

00:01:13.995 --> 00:01:17.037
write that, maybe this
is equal to infinity.

00:01:17.037 --> 00:01:19.199
But infinity isn't, we're not talking

00:01:19.199 --> 00:01:20.828
about a specific number.

00:01:20.828 --> 00:01:23.957
If we're talking technically about limits

00:01:23.957 --> 00:01:26.888
the way that we've looked at it, what is,

00:01:26.888 --> 00:01:29.285
you'll sometimes see this in some classes.

00:01:29.285 --> 00:01:31.187
But in this context,
especially on the exercises

00:01:31.187 --> 00:01:34.845
on Khan Academy, we'll say
that this does not exist.

00:01:34.845 --> 00:01:35.678
Not exist.

00:01:36.783 --> 00:01:39.327
This thing right over here is unbounded.

00:01:39.327 --> 00:01:41.129
Now this is interesting because
the left-handed limit here

00:01:41.129 --> 00:01:43.944
doesn't exist, but the
right-handed limit does.

00:01:43.944 --> 00:01:46.876
If I were to say the limit of g of x

00:01:46.876 --> 00:01:51.577
as x approaches six from
the right-hand side,

00:01:51.577 --> 00:01:52.786
well, let's see.

00:01:52.786 --> 00:01:56.313
We have g of eight is
there, go of seven is there,

00:01:56.313 --> 00:01:58.396
g of 6.5, looks like it's

00:01:59.336 --> 00:02:01.665
a little less than negative three.

00:02:01.665 --> 00:02:05.280
G of 6.01, little even
closer to negative three.

00:02:05.280 --> 00:02:09.197
G of 6.0000001 is very
close to negative three.

00:02:12.487 --> 00:02:14.601
So it looks like this
limit right over here,

00:02:14.601 --> 00:02:16.115
at least looking at it graphically,

00:02:16.115 --> 00:02:18.608
it looks like when we approach six

00:02:18.608 --> 00:02:20.495
from the right, looks like the function

00:02:20.495 --> 00:02:21.855
is approaching negative three.

00:02:21.855 --> 00:02:23.787
But from the left, it's just unbounded,

00:02:23.787 --> 00:02:25.864
so we'll say it doesn't exist.


Graphical limit at asymptotic discontinuity

WEBVTT
Kind: captions
Language: en

00:00:00.115 --> 00:00:01.744
- [Voiceover] Alright we have a graph

00:00:01.744 --> 00:00:03.299
of y is equal to f of x

00:00:03.299 --> 00:00:05.125
and we wanna figure out
what is the limit of f of x

00:00:05.125 --> 00:00:07.836
as x approaches negative three.

00:00:07.836 --> 00:00:11.173
And if we just look at
x equals negative three,

00:00:11.173 --> 00:00:12.512
it's really hard to see

00:00:12.512 --> 00:00:14.652
at least based on how this graph looks,

00:00:14.652 --> 00:00:19.419
how what f of negative
three is and if anything,

00:00:19.419 --> 00:00:23.700
it looks like we have an
asymptotic discontinuity here.

00:00:23.700 --> 00:00:27.617
It looks like on the left side
of x equals negative three,

00:00:27.617 --> 00:00:29.479
it looks like we are approaching,

00:00:29.479 --> 00:00:30.607
I guess you can say infinity

00:00:30.607 --> 00:00:31.619
and on the right side,

00:00:31.619 --> 00:00:33.804
it looks like we are
approaching infinity as well.

00:00:33.804 --> 00:00:34.778
And we can just look at that and say,

00:00:34.778 --> 00:00:36.778
What is f of negative 5?

00:00:38.054 --> 00:00:39.062
That's four.

00:00:39.062 --> 00:00:41.751
F of negative four is
looks like it is around,

00:00:41.751 --> 00:00:43.009
I don't know around eight.

00:00:43.009 --> 00:00:45.950
F of negative three is off the charts.

00:00:45.950 --> 00:00:46.949
F of negative,

00:00:46.949 --> 00:00:49.811
if we continued with this trend
and if we were to asymptote

00:00:49.811 --> 00:00:51.589
towards this line right over here,

00:00:51.589 --> 00:00:53.063
this vertical asymptote,

00:00:53.063 --> 00:00:56.081
it looks like as we get closer
and closer to negative three

00:00:56.081 --> 00:01:00.230
that the value of the function
at that point is approaching,

00:01:00.230 --> 00:01:02.127
is getting closer and closer to infinity,

00:01:02.127 --> 00:01:03.616
at least that's what it looks like

00:01:03.616 --> 00:01:05.220
from what we can see on this graph

00:01:05.220 --> 00:01:07.988
as we approach negative three
from the left hand side.

00:01:07.988 --> 00:01:10.112
Let's think about as we
approach negative three

00:01:10.112 --> 00:01:11.557
from the right hand side,

00:01:11.557 --> 00:01:14.571
so this is f of negative one,

00:01:14.571 --> 00:01:15.804
f of negative two,

00:01:15.804 --> 00:01:20.340
f of negative 2.5 looks like
it's up here some place,

00:01:20.340 --> 00:01:24.362
f of negative 2.9 would be even higher,

00:01:24.362 --> 00:01:26.746
f of negative 2.999
looks like it will just

00:01:26.746 --> 00:01:28.847
once again approach infinity

00:01:28.847 --> 00:01:32.581
so this type of limit in some context

00:01:32.581 --> 00:01:35.189
you'd say this limit doesn't exist

00:01:35.189 --> 00:01:36.022
doesn't

00:01:37.193 --> 00:01:40.462
doesn't exist in the formal sense

00:01:40.462 --> 00:01:42.034
so that's one way to think about it.

00:01:42.034 --> 00:01:44.197
In some contexts,

00:01:44.197 --> 00:01:47.554
you'll hear people say that this limit

00:01:47.554 --> 00:01:49.498
since from the left and from the right,

00:01:49.498 --> 00:01:50.517
it looks like it's going to infinity.

00:01:50.517 --> 00:01:52.775
Sometimes you'll see people say

00:01:52.775 --> 00:01:54.869
that it's approaching infinity

00:01:54.869 --> 00:01:57.248
and so this is depending
on what type of a class

00:01:57.248 --> 00:01:58.802
what type of a context you're in

00:01:58.802 --> 00:02:01.470
but in the traditional sense of the limit

00:02:01.470 --> 00:02:03.014
or in the technical sense,

00:02:03.014 --> 00:02:04.986
there are ways that you can define limits

00:02:04.986 --> 00:02:07.041
where this will make a
little bit more sense

00:02:07.041 --> 00:02:10.646
but the traditional definition of a limit,

00:02:10.646 --> 00:02:11.637
it would be,

00:02:11.637 --> 00:02:15.387
you would say that this
limit does not exist.


Graphical limit at point discontinuity

WEBVTT
Kind: captions
Language: en

00:00:00.222 --> 00:00:01.641
- [Voiceover] So, here we have the graph

00:00:01.641 --> 00:00:03.391
y is equal to g of x.

00:00:04.754 --> 00:00:07.919
We have a little point
discontinuity right over here

00:00:07.919 --> 00:00:10.688
at x is equal to seven,

00:00:10.688 --> 00:00:12.361
and what we want to do is figure out

00:00:12.361 --> 00:00:17.244
what is the limit of g of
x as x approaches seven.

00:00:17.244 --> 00:00:20.199
So, since you say, well, what
is the function approaching

00:00:20.199 --> 00:00:23.123
as the inputs in the function
are approaching seven?

00:00:23.123 --> 00:00:26.000
So, let's see, so if we input
as the input to the function

00:00:26.000 --> 00:00:28.785
approaches seven from
values less than seven,

00:00:28.785 --> 00:00:32.189
so if x is three, g of three is here,

00:00:32.189 --> 00:00:36.448
g of three is right there,
g of four is right there,

00:00:36.448 --> 00:00:39.281
g of five is right there, g of six

00:00:40.337 --> 00:00:42.820
looks like it's a little bit more than,

00:00:42.820 --> 00:00:46.055
or a little bit less than negative one,

00:00:46.055 --> 00:00:49.888
g of 6.5 looks like it's
around negative half,

00:00:52.306 --> 00:00:55.473
g of negative 6.9 is right over there,

00:00:56.458 --> 00:01:00.233
looks like it's a little
bit less than zero,

00:01:00.233 --> 00:01:03.939
g of negative 6.999 looks
like it's a little bit,

00:01:03.939 --> 00:01:05.686
it's still less than zero
but it's a little bit

00:01:05.686 --> 00:01:08.889
closer to zero so it looks
like we're getting closer

00:01:08.889 --> 00:01:13.378
as x gets closer and closer
but not quite at seven,

00:01:13.378 --> 00:01:14.867
it looks like the value of our function

00:01:14.867 --> 00:01:17.231
is approaching zero.

00:01:17.231 --> 00:01:19.234
Let's see if that's also true from values

00:01:19.234 --> 00:01:21.127
for x values greater than seven,

00:01:21.127 --> 00:01:24.238
so g of nine is up here,
looks like it's around six,

00:01:24.238 --> 00:01:26.823
g of eight looks like it's
a little bit more than two,

00:01:26.823 --> 00:01:29.743
g of 7.5 looks like it's a
little bit more than one,

00:01:29.743 --> 00:01:34.597
g of 7.1 looks like it's a
little bit more than zero,

00:01:34.597 --> 00:01:37.638
the g of 7.1 looks like it's
a little bit more than zero,

00:01:37.638 --> 00:01:41.093
g of 7.01 is even closer to zero,

00:01:41.093 --> 00:01:45.260
g of 7.0000001 looks like
it'll be even closer to zero

00:01:46.144 --> 00:01:48.786
so once again it looks like
we are approaching zero

00:01:48.786 --> 00:01:50.236
as x as approaches seven,

00:01:50.236 --> 00:01:53.055
in this case as we approach
from larger values of seven.

00:01:53.055 --> 00:01:55.744
And this is interesting because the limit

00:01:55.744 --> 00:01:58.929
as x approaches seven of
g of x is different than

00:01:58.929 --> 00:02:01.980
the function's actual value, g of seven,

00:02:01.980 --> 00:02:04.245
when we actually input
seven into the function.

00:02:04.245 --> 00:02:06.024
When we actually input
seven into the function,

00:02:06.024 --> 00:02:08.955
we can see the graph tells us
that the value of the function

00:02:08.955 --> 00:02:11.210
is equal to three.

00:02:11.210 --> 00:02:13.562
So we actually have this
point discontinuity,

00:02:13.562 --> 00:02:16.034
or sometimes called a
removable discontinuity,

00:02:16.034 --> 00:02:17.813
right over here, and this is,

00:02:17.813 --> 00:02:19.550
I'm not gonna do a lot of depth here,

00:02:19.550 --> 00:02:22.321
but this is starting to touch on how we,

00:02:22.321 --> 00:02:24.807
one of the ways that we can
actually test for continuity

00:02:24.807 --> 00:02:27.598
is if the limit as we approach
a value is not the same

00:02:27.598 --> 00:02:30.785
as the actual value of the
function of that point,

00:02:30.785 --> 00:02:33.294
well, then we're probably talking about,

00:02:33.294 --> 00:02:37.461
or actually we are talking
about, a discontinuity.


Graphical limit where function undefined

WEBVTT
Kind: captions
Language: en

00:00:00.256 --> 00:00:01.565
- [Voiceover] So we have the graph

00:00:01.565 --> 00:00:04.676
of y equal f of x right over here,

00:00:04.676 --> 00:00:06.170
and what we want to do is figure out

00:00:06.170 --> 00:00:10.273
the limit of f of x as x
approaches negative four.

00:00:10.273 --> 00:00:12.132
So what does that mean?

00:00:12.132 --> 00:00:16.778
Well, a limit is saying, "What
does my function approach

00:00:16.778 --> 00:00:19.378
"as the input of that function approaches,

00:00:19.378 --> 00:00:20.817
"in this case, negative four?

00:00:20.817 --> 00:00:23.175
"As the input approaches a value?"

00:00:23.175 --> 00:00:24.436
And as we see in this example,

00:00:24.436 --> 00:00:25.450
the function doesn't necessarily

00:00:25.450 --> 00:00:27.796
have to be even defined at that value.

00:00:27.796 --> 00:00:30.067
We can see f of negative four,

00:00:30.067 --> 00:00:32.416
you go to x equals negative four,

00:00:32.416 --> 00:00:35.955
and you see that f of
negative four is undefined.

00:00:35.955 --> 00:00:38.756
So this is not defined, but as we'll see

00:00:38.756 --> 00:00:40.810
even though the function
isn't defined there,

00:00:40.810 --> 00:00:43.476
the limit might be defined there.

00:00:43.476 --> 00:00:45.264
And actually it can go
the other way around.

00:00:45.264 --> 00:00:48.252
Sometimes a function is defined there,

00:00:48.252 --> 00:00:52.381
but the limit is not, and we'll
see that in future videos.

00:00:52.381 --> 00:00:53.923
But let's just get an understanding here.

00:00:53.923 --> 00:00:57.072
What's going on as x
approaches negative four?

00:00:57.072 --> 00:00:59.204
As x approaches negative four

00:00:59.204 --> 00:01:01.580
from values greater than negative four

00:01:01.580 --> 00:01:04.785
and from values less than negative four.

00:01:04.785 --> 00:01:05.880
Well let's first think about values

00:01:05.880 --> 00:01:07.455
greater than negative four.

00:01:07.455 --> 00:01:11.482
So when x is negative one,
this is f of negative one.

00:01:11.482 --> 00:01:15.760
This is f of negative two,
this is f of negative three.

00:01:15.760 --> 00:01:17.927
This is f of negative 3.5,

00:01:20.218 --> 00:01:22.737
this is f of negative 3.9,

00:01:22.737 --> 00:01:27.451
this is f of negative 3.99,
this is f of negative 3.99999.

00:01:27.451 --> 00:01:29.874
And so you can see the
value of our function,

00:01:29.874 --> 00:01:32.528
as x gets closer and
closer to negative four

00:01:32.528 --> 00:01:34.065
from values greater than negative four,

00:01:34.065 --> 00:01:36.398
seems to be approaching six.

00:01:37.522 --> 00:01:39.077
And let's see if that's true
from the other direction,

00:01:39.077 --> 00:01:41.848
so from values less than negative four.

00:01:41.848 --> 00:01:45.765
So this f of negative
seven, f of negative six.

00:01:46.925 --> 00:01:51.008
This is f of negative
five, puts us around seven.

00:01:52.448 --> 00:01:56.198
This is f of negative
4.5, f of negative 4.1,

00:01:58.767 --> 00:02:00.350
f of negative 4.01,

00:02:01.492 --> 00:02:03.325
f of negative 4.00001,

00:02:05.822 --> 00:02:06.952
it looks like it's getting awfully,

00:02:06.952 --> 00:02:08.422
it's going to be a
little bit more than six,

00:02:08.422 --> 00:02:11.432
so it seems as we get closer and closer

00:02:11.432 --> 00:02:14.062
to x equal negative four, the value

00:02:14.062 --> 00:02:17.729
of our function is
approaching positive six.


Introduction to "Meet a chemistry professional"

WEBVTT
Kind: captions
Language: en

00:00:01.423 --> 00:00:02.718
- [Voiceover] Have you ever wondered

00:00:02.718 --> 00:00:04.968
what a chemist really does?

00:00:04.968 --> 00:00:07.759
In this series we asked people
with chemistry backgrounds

00:00:07.759 --> 00:00:09.277
to share their stories.

00:00:09.277 --> 00:00:12.884
We have people from all
different fields and careers.

00:00:12.884 --> 00:00:15.596
For example, we have
an interview of someone

00:00:15.596 --> 00:00:18.047
who works in forensics,

00:00:18.047 --> 00:00:22.548
which is closely related
to analytical chemistry.

00:00:22.548 --> 00:00:24.548
We also have people from

00:00:28.003 --> 00:00:30.753
the field of science illustration

00:00:33.169 --> 00:00:34.252
and medicine.

00:00:36.034 --> 00:00:39.536
Since if you've ever
thought about being pre-med,

00:00:39.536 --> 00:00:41.721
or doing anything else
related to medicine,

00:00:41.721 --> 00:00:43.541
you probably know by now
that you'll be taking

00:00:43.541 --> 00:00:46.139
a fair number of chemistry classes.

00:00:46.139 --> 00:00:49.357
We have someone who works
in industry as a researcher

00:00:49.357 --> 00:00:51.357
developing new products.

00:00:53.644 --> 00:00:57.477
We also have someone who
works as a programmer

00:01:00.416 --> 00:01:02.499
at Khan Academy actually.

00:01:03.581 --> 00:01:06.535
We have a bunch of other
interviews for careers

00:01:06.535 --> 00:01:08.917
where it was a little
harder to find pictures,

00:01:08.917 --> 00:01:11.184
but they're all really cool jobs.

00:01:11.184 --> 00:01:13.767
We have fields that are covered

00:01:15.353 --> 00:01:18.436
include patent law and science media,

00:01:21.026 --> 00:01:24.943
which includes things
like making and producing

00:01:26.575 --> 00:01:28.992
and hosting science TV shows.

00:01:30.228 --> 00:01:32.145
We have an engineer who

00:01:35.367 --> 00:01:37.102
was not a chemistry major

00:01:37.102 --> 00:01:38.885
but had to take some chemistry classes,

00:01:38.885 --> 00:01:42.218
and ended up using chemistry in her job.

00:01:43.689 --> 00:01:46.355
We have an epidemiologist,

00:01:46.355 --> 00:01:50.355
which is a really hard
word to spell, miologist.

00:01:54.643 --> 00:01:58.643
It's also a really
interesting career, and more.

00:01:59.860 --> 00:02:02.661
We hope to continue
adding to this collection

00:02:02.661 --> 00:02:05.637
of interviews over time,
so please check back.

00:02:05.637 --> 00:02:07.707
But hopefully you can tell from this list

00:02:07.707 --> 00:02:10.789
that all of these people
who studied chemistry

00:02:10.789 --> 00:02:12.274
are doing a lot of different things.

00:02:12.274 --> 00:02:14.107
So there's a lot of ways
you can use chemistry

00:02:14.107 --> 00:02:16.524
as part of your career after school.

00:02:16.524 --> 00:02:19.538
We hope you find these interviews
interesting and helpful

00:02:19.538 --> 00:02:23.705
in figuring out what you
might want to do as a career.


Uncle Tom's Cabin part 3

WEBVTT
Kind: captions
Language: en

00:00:00.746 --> 00:00:01.579
- [Voiceover] Hey Kim.

00:00:01.579 --> 00:00:02.605
- [Kim] Hey, Becca.

00:00:02.605 --> 00:00:05.233
So, we've been talking
about Uncle Tom's Cabin,

00:00:05.233 --> 00:00:08.592
published in 1852 by
Harriet Beecher Stowe,

00:00:08.592 --> 00:00:10.724
and said to have been
one of the main causes

00:00:10.724 --> 00:00:13.082
of the American Civil War.

00:00:13.082 --> 00:00:15.391
So, remind me again of
what Uncle Tom's Cabin

00:00:15.391 --> 00:00:16.798
was actually about.

00:00:16.798 --> 00:00:18.935
- [Becca] So Uncle Tom's Cabin was about

00:00:18.935 --> 00:00:21.841
the horrors of slavery in the deep south

00:00:21.841 --> 00:00:25.996
and also appealed to a lot of
American's Christian values

00:00:25.996 --> 00:00:28.657
and tried to point out these
fundamental contradictions

00:00:28.657 --> 00:00:31.314
between Christian faith and slavery.

00:00:31.314 --> 00:00:33.081
- [Kim] So, it was published in 1852.

00:00:33.081 --> 00:00:34.764
How did people receive this book?

00:00:34.764 --> 00:00:36.980
- [Becca] So there was a
lot of mixed reactions.

00:00:36.980 --> 00:00:40.710
But, it was the most read
book of the 19th century,

00:00:40.710 --> 00:00:42.889
so there were a lot of reactions.

00:00:42.889 --> 00:00:43.722
(laughing)

00:00:43.722 --> 00:00:47.785
In the north, mostly, people
were reading this all the time,

00:00:47.785 --> 00:00:50.205
no matter if you were an intellectual

00:00:50.205 --> 00:00:54.056
or just kind of a 17 year old
picking up a book off a shelf,

00:00:54.056 --> 00:00:55.864
you were going to read Uncle Tom's Cabin.

00:00:55.864 --> 00:00:57.655
And a lot of the people
that actually read the book

00:00:57.655 --> 00:01:01.503
were young men that would
later fight in the Civil War.

00:01:01.503 --> 00:01:02.336
- [Kim] How interesting.

00:01:02.336 --> 00:01:05.279
Okay, so, like this would be
our book club book of the month

00:01:05.279 --> 00:01:06.553
except everybody was,

00:01:06.553 --> 00:01:11.046
this was like the Oprah's
book club choice of 1852.

00:01:11.046 --> 00:01:12.245
- [Becca] Exactly.
- [Kim] Everybody was

00:01:12.245 --> 00:01:13.227
reading it.

00:01:13.227 --> 00:01:15.773
- [Becca] Yeah, there's
really almost nothing like it.

00:01:15.773 --> 00:01:17.212
Not even Harry Potter.

00:01:17.212 --> 00:01:19.199
But it did have the
same international scope

00:01:19.199 --> 00:01:20.844
that Harry Potter does today.

00:01:20.844 --> 00:01:23.605
- [Kim] So, it was popular in
the northern United States,

00:01:23.605 --> 00:01:25.215
and elsewhere in the world.

00:01:25.215 --> 00:01:26.655
So, where else was it popular?

00:01:26.655 --> 00:01:28.972
- [Becca] Mostly in Europe.

00:01:28.972 --> 00:01:31.322
But it was translated
into over 60 languages.

00:01:31.322 --> 00:01:32.191
- [Kim] Wow.

00:01:32.191 --> 00:01:34.589
- [Becca] And, this also
kind of put the spotlight

00:01:34.589 --> 00:01:36.773
on American slavery.

00:01:36.773 --> 00:01:39.782
So, there was all this
international attention.

00:01:39.782 --> 00:01:41.304
What is going on in America,

00:01:41.304 --> 00:01:42.576
and what's going to happen?

00:01:42.576 --> 00:01:43.409
- [Kim] That's so interesting.

00:01:43.409 --> 00:01:45.170
It reminds me of The King and I.

00:01:45.170 --> 00:01:49.377
If you've seen that she,
the woman goes to Siam,

00:01:49.377 --> 00:01:50.794
and shows people

00:01:52.093 --> 00:01:54.103
the book Uncle Tom's Cabin

00:01:54.103 --> 00:01:56.898
and they put on a version of a play

00:01:56.898 --> 00:01:58.295
based on Uncle Tom's Cabin

00:01:58.295 --> 00:02:00.515
in what would be Thailand.

00:02:00.515 --> 00:02:02.032
- [Becca] So yeah, so
this makes it this kind of

00:02:02.032 --> 00:02:04.210
international spectacle.

00:02:04.210 --> 00:02:07.240
The fate of slavery had
to be somehow figured out,

00:02:07.240 --> 00:02:09.256
and everyone was watching.

00:02:09.256 --> 00:02:10.089
- [Kim] That's so interesting.

00:02:10.089 --> 00:02:12.165
So, I'm imagining that white southerners

00:02:12.165 --> 00:02:14.135
were not big fans of this book.

00:02:14.135 --> 00:02:15.938
- [Becca] So, white
southerners were definitely

00:02:15.938 --> 00:02:18.682
not a fan of Uncle Tom's Cabin.

00:02:18.682 --> 00:02:21.098
And in response, there was
this movement of these things

00:02:21.098 --> 00:02:23.186
called anti-Tom novels.

00:02:23.186 --> 00:02:25.976
Here's an anti-Tom novel right next to us,

00:02:25.976 --> 00:02:26.919
right over here,

00:02:26.919 --> 00:02:28.016
Aunt Phillis's Cabin.

00:02:28.016 --> 00:02:28.849
(Kim laughing)

00:02:28.849 --> 00:02:30.900
- [Kim] Creative name.
- [Becca] Aunt Phillis' Cabin,

00:02:30.900 --> 00:02:32.353
yes (laughing), very creative name.

00:02:32.353 --> 00:02:35.269
These anti-Tom novels
aimed to point out that

00:02:35.269 --> 00:02:36.997
maybe Harriet Beecher Stowe didn't know

00:02:36.997 --> 00:02:38.521
what she was actually talking about.

00:02:38.521 --> 00:02:40.246
They also accused Harriet Beecher Stowe

00:02:40.246 --> 00:02:42.758
of not actually even
living in the deep south,

00:02:42.758 --> 00:02:44.991
so she didn't even know
what slavery was like.

00:02:44.991 --> 00:02:47.781
They wanted to paint
southern slave society

00:02:47.781 --> 00:02:49.378
in this really positive light.

00:02:49.378 --> 00:02:51.591
They wanted to show all
the ways that it actually

00:02:51.591 --> 00:02:53.276
maintains social order,

00:02:53.276 --> 00:02:55.573
and promoted economic welfare,

00:02:55.573 --> 00:02:58.115
so this was kind of this
response from the south,

00:02:58.115 --> 00:02:59.630
also in novel form.

00:02:59.630 --> 00:03:01.885
- [Kim] So, it's this big
kind of cultural battle

00:03:01.885 --> 00:03:04.029
over the interpretation of slavery.

00:03:04.029 --> 00:03:05.918
You have people on one hand saying

00:03:05.918 --> 00:03:08.256
slavery is destructive to families,

00:03:08.256 --> 00:03:11.449
slavery is incompatible with Christianity.

00:03:11.449 --> 00:03:14.243
And then, responses from the white south,

00:03:14.243 --> 00:03:16.702
saying, oh no, actually slavery is great,

00:03:16.702 --> 00:03:18.185
it helps everybody.

00:03:18.185 --> 00:03:20.035
- [Becca] Right, so there
was this kind of battle

00:03:20.035 --> 00:03:22.010
within the literary community

00:03:22.010 --> 00:03:24.637
about the peculiar institution of slavery.

00:03:24.637 --> 00:03:25.625
Which one was it?

00:03:25.625 --> 00:03:28.414
- [Kim] So what if, I'm illiterate, right.

00:03:28.414 --> 00:03:31.162
I mean, not everybody
in 19th century America

00:03:31.162 --> 00:03:33.848
was a New England intellectual

00:03:33.848 --> 00:03:37.014
who was reading Christian novels.

00:03:37.014 --> 00:03:39.027
How would I have heard
about Uncle Tom's Cabin?

00:03:39.027 --> 00:03:40.636
- [Becca] That's a really good point, Kim.

00:03:40.636 --> 00:03:44.778
So, Tom Shows were depictions
of Uncle Tom's Cabin

00:03:44.778 --> 00:03:46.337
in theaters around the world.

00:03:46.337 --> 00:03:49.993
And so, they were often times
put on by abolitionist people

00:03:49.993 --> 00:03:53.648
trying to point out the
issues with slavery today

00:03:53.648 --> 00:03:56.229
and end slavery immediately.

00:03:56.229 --> 00:03:58.196
- [Kim] So, this is before
there's copyright law, right,

00:03:58.196 --> 00:04:01.157
so you can just put on a
show of anybody's novel

00:04:01.157 --> 00:04:02.396
if you feel like it.

00:04:02.396 --> 00:04:03.993
- [Becca] Exactly, but often times,

00:04:03.993 --> 00:04:06.490
they really misconstrued the novel.

00:04:06.490 --> 00:04:09.491
And actually, now are
remembered as contributing

00:04:09.491 --> 00:04:12.684
to the problem of racism in America.

00:04:12.684 --> 00:04:14.493
- [Kim] Racial stereotypes,
too, I would imagine,

00:04:14.493 --> 00:04:17.201
because we still have
this phrase Uncle Tom

00:04:17.201 --> 00:04:18.929
kind of to mean

00:04:18.929 --> 00:04:23.689
an African-American who is
a martyr to the status quo

00:04:23.689 --> 00:04:26.778
as opposed to someone who
might fight against racism.

00:04:26.778 --> 00:04:28.258
It seems like they might have borrowed

00:04:28.258 --> 00:04:30.596
a lot of these stereotypes
from minstrel shows

00:04:30.596 --> 00:04:33.142
which were also very
popular in this time period.

00:04:33.142 --> 00:04:34.340
- [Becca] And some of the characters

00:04:34.340 --> 00:04:35.654
within these minstrel shows

00:04:35.654 --> 00:04:39.304
turned into the character that
was remembered as Jim Crow,

00:04:39.304 --> 00:04:41.483
which became the dominating racial order

00:04:41.483 --> 00:04:43.695
after the Civil War.

00:04:43.695 --> 00:04:47.010
So, in the Civil Rights
era in the mid-1950's,

00:04:47.010 --> 00:04:49.761
lots of activists actually
wanted to completely reject

00:04:49.761 --> 00:04:52.931
the progress that Uncle Tom's
Cabin and these Tom Shows

00:04:52.931 --> 00:04:55.808
had made because they actually
reduced African Americans

00:04:55.808 --> 00:04:58.065
to this terrible stereotype.

00:04:58.065 --> 00:05:01.233
And so, later on, this kind of idea that

00:05:01.233 --> 00:05:04.970
someone was an Uncle Tom
became a racial slur, really,

00:05:04.970 --> 00:05:07.919
and they then rejected Uncle Tom's Cabin

00:05:07.919 --> 00:05:11.124
as being a tool towards racial equality

00:05:11.124 --> 00:05:14.170
and more saw it as a part of the problem.

00:05:14.170 --> 00:05:16.228
- [Kim] So, I think,
the most important thing

00:05:16.228 --> 00:05:19.393
about Uncle Tom's Cabin
is that it's this catalyst

00:05:19.393 --> 00:05:22.726
of really intense emotions about slavery

00:05:24.032 --> 00:05:28.300
which in the 1850's will lead
eventually to the Civil War.

00:05:28.300 --> 00:05:31.750
And, following the publication
of Uncle Tom's Cabin,

00:05:31.750 --> 00:05:33.848
you'll see more and more violence

00:05:33.848 --> 00:05:35.660
on both sides of this issue.

00:05:35.660 --> 00:05:38.532
For example, John Brown,
this famous abolitionist

00:05:38.532 --> 00:05:41.942
actually goes out to
Kansas and murders people.

00:05:41.942 --> 00:05:44.287
- [Becca] And I think that
Lincoln was very astute

00:05:44.287 --> 00:05:46.015
in pointing out that Uncle Tom's Cabin

00:05:46.015 --> 00:05:49.294
really catalyzed a lot of this violence.

00:05:49.294 --> 00:05:51.313
And he even met with
Harriet Beecher Stowe,

00:05:51.313 --> 00:05:54.479
so she earned herself
a little meeting with

00:05:54.479 --> 00:05:55.339
Abraham Lincoln.

00:05:55.339 --> 00:05:56.197
(Kim laughing)

00:05:56.197 --> 00:05:57.109
- [Kim] And he said,

00:05:57.109 --> 00:06:01.276
"So you're the little lady
that started this great war."

00:06:03.549 --> 00:06:07.819
I'm trying to think of another
book that has started a war.

00:06:07.819 --> 00:06:10.728
- [Becca] I think we would
probably remember that.

00:06:10.728 --> 00:06:13.152
But I do think Lincoln was
really astute in pointing out

00:06:13.152 --> 00:06:16.200
just how impactful this
cultural phenomenon,

00:06:16.200 --> 00:06:19.655
this Tom-mania was on
the question of slavery

00:06:19.655 --> 00:06:22.641
and on the fate of the American people.

00:06:22.641 --> 00:06:25.682
And really, it just begged
the question in a new way,

00:06:25.682 --> 00:06:28.553
in this kind of public setting.

00:06:28.553 --> 00:06:30.516
I mean, I just think that the book itself,

00:06:30.516 --> 00:06:32.485
the way that the book could just travel

00:06:32.485 --> 00:06:33.962
all around the United States

00:06:33.962 --> 00:06:35.607
and so many different kinds of people

00:06:35.607 --> 00:06:37.981
were able to read it and
get their hands on it.

00:06:37.981 --> 00:06:40.764
This really was just
this movement of people

00:06:40.764 --> 00:06:42.366
just thinking a lot about slavery,

00:06:42.366 --> 00:06:43.842
reading a lot about slavery.

00:06:43.842 --> 00:06:47.162
- [Kim] Yeah, well, I think
after Uncle Tom's Cabin,

00:06:47.162 --> 00:06:49.792
I don't think there was a
way to not have an opinion

00:06:49.792 --> 00:06:51.843
on the slavery issue.

00:06:51.843 --> 00:06:53.075
Either you were for it,

00:06:53.075 --> 00:06:54.395
or you were against it.

00:06:54.395 --> 00:06:57.303
And that divisiveness would
lead to the Civil War.

00:06:57.303 --> 00:06:59.677
- [Becca] And again, there's
this international focus.

00:06:59.677 --> 00:07:01.969
There's a deeper, sectional divide

00:07:01.969 --> 00:07:03.654
between the north and the south,

00:07:03.654 --> 00:07:06.434
and there was this kind of
sorting over the slavery question

00:07:06.434 --> 00:07:08.311
that Uncle Tom's Cabin really promoted.