Limits at infinity of quotients (Part 2)

WEBVTT
Kind: captions
Language: en

00:00:00.550 --> 00:00:03.540
Let's do a few more examples of
finding the limit of functions

00:00:03.540 --> 00:00:07.020
as x approaches infinity
or negative infinity.

00:00:07.020 --> 00:00:08.870
So here I have this
crazy function.

00:00:08.870 --> 00:00:11.110
9x to the seventh
minus 17x to the sixth,

00:00:11.110 --> 00:00:12.340
plus 15 square roots of x.

00:00:12.340 --> 00:00:14.540
All of that over 3x to
the seventh plus 1,000x

00:00:14.540 --> 00:00:17.320
to the fifth, minus
log base 2 of x.

00:00:17.320 --> 00:00:20.500
So what's going to happen
as x approaches infinity?

00:00:20.500 --> 00:00:22.810
And the key here, like we've
seen in other examples,

00:00:22.810 --> 00:00:26.410
is just to realize which
terms will dominate.

00:00:26.410 --> 00:00:27.940
So for example,
in the numerator,

00:00:27.940 --> 00:00:30.680
out of these three terms,
the 9x to the seventh

00:00:30.680 --> 00:00:34.660
is going to grow much faster
than any of these other terms.

00:00:34.660 --> 00:00:38.460
So this is the dominating
term in the numerator.

00:00:38.460 --> 00:00:40.800
And in the denominator,
3x to the seventh

00:00:40.800 --> 00:00:43.220
is going to grow much faster
than an x to the fifth term,

00:00:43.220 --> 00:00:47.190
and definitely much faster
than a log base 2 term.

00:00:47.190 --> 00:00:50.150
So at infinity, as we get
closer and closer to infinity,

00:00:50.150 --> 00:00:54.250
this function is going
to be roughly equal to 9x

00:00:54.250 --> 00:00:58.070
to the seventh over
3x to the seventh.

00:00:58.070 --> 00:00:59.899
And so we can say,
especially since,

00:00:59.899 --> 00:01:02.190
as we get larger and larger
as we get closer and closer

00:01:02.190 --> 00:01:04.360
to infinity, these
two things are

00:01:04.360 --> 00:01:06.347
going to get closer
and closer each other.

00:01:06.347 --> 00:01:07.930
We could say this
limit is going to be

00:01:07.930 --> 00:01:10.850
the same thing as this limit.

00:01:10.850 --> 00:01:12.940
Which is going to be
equal to the limit

00:01:12.940 --> 00:01:15.340
as x approaches infinity.

00:01:15.340 --> 00:01:17.640
Well, we can just cancel
out the x to the seventh.

00:01:17.640 --> 00:01:20.380
So it's going to
be 9/3, or just 3.

00:01:20.380 --> 00:01:22.460
Which is just going to be 3.

00:01:22.460 --> 00:01:24.740
So that is our limit, as
x approaches infinity,

00:01:24.740 --> 00:01:26.646
in all of this craziness.

00:01:26.646 --> 00:01:28.770
Now let's do the same with
this function over here.

00:01:28.770 --> 00:01:30.275
Once again, crazy function.

00:01:30.275 --> 00:01:31.650
We're going to
negative infinity.

00:01:31.650 --> 00:01:33.310
But the same principles apply.

00:01:33.310 --> 00:01:36.940
Which terms dominate as
the absolute value of x

00:01:36.940 --> 00:01:38.420
get larger and
larger and larger?

00:01:38.420 --> 00:01:40.960
As x gets larger in magnitude.

00:01:40.960 --> 00:01:43.820
Well, in the numerator, it's
the 3x to the third term.

00:01:43.820 --> 00:01:47.020
In the denominator it's
the 6x to the fourth term.

00:01:47.020 --> 00:01:50.730
So this is going to be the
same thing as the limit of 3x

00:01:50.730 --> 00:01:54.940
to the third over 6x to
the fourth, as x approaches

00:01:54.940 --> 00:01:56.304
negative infinity.

00:01:56.304 --> 00:01:57.720
And if we simplified
this, this is

00:01:57.720 --> 00:02:00.950
going to be equal to the
limit as x approaches

00:02:00.950 --> 00:02:05.850
negative infinity of 1 over 2x.

00:02:05.850 --> 00:02:07.900
And what's this going to be?

00:02:07.900 --> 00:02:09.800
Well, if the denominator,
even though it's

00:02:09.800 --> 00:02:12.510
becoming a larger and larger
and larger negative number,

00:02:12.510 --> 00:02:16.700
it becomes 1 over a very,
very large negative number.

00:02:16.700 --> 00:02:18.930
Which is going to get us
pretty darn close to 0.

00:02:18.930 --> 00:02:21.850
Just as 1 over x, as x
approaches negative infinity,

00:02:21.850 --> 00:02:23.490
gets us close to 0.

00:02:23.490 --> 00:02:25.820
So this right over here,
the horizontal asymptote

00:02:25.820 --> 00:02:28.362
in this case, is
y is equal to 0.

00:02:28.362 --> 00:02:30.820
And I encourage you to graph
it, or try it out with numbers

00:02:30.820 --> 00:02:32.920
to verify that for yourself.

00:02:32.920 --> 00:02:36.540
The key realization here
is to simplify the problem

00:02:36.540 --> 00:02:38.100
by just thinking
about which terms

00:02:38.100 --> 00:02:41.979
are going to dominate the rest.

00:02:41.979 --> 00:02:43.270
Now let's think about this one.

00:02:43.270 --> 00:02:45.240
What is the limit of
this crazy function

00:02:45.240 --> 00:02:47.520
as x approaches infinity?

00:02:47.520 --> 00:02:49.645
Well, once again, what
are the dominating terms?

00:02:49.645 --> 00:02:52.270
In the numerator, it's 4x to the
fourth, and in the denominator

00:02:52.270 --> 00:02:54.500
it's 250x to the third.

00:02:54.500 --> 00:02:56.280
These are the
highest degree terms.

00:02:56.280 --> 00:02:58.720
So this is going to be the
same thing as the limit,

00:02:58.720 --> 00:03:08.760
as x approaches infinity, of
4x to the fourth over 250x

00:03:08.760 --> 00:03:09.920
to the third.

00:03:09.920 --> 00:03:12.880
Which is going to be the same
thing as the limit of-- let's

00:03:12.880 --> 00:03:14.920
see, 4, well I
could just-- this is

00:03:14.920 --> 00:03:17.635
going to be the same thing
as-- well we could divide two hundred

00:03:17.635 --> 00:03:19.460
and, well, I'll just
leave it like this.

00:03:19.460 --> 00:03:23.120
It's going to be the
limit of 4 over 250.

00:03:23.120 --> 00:03:25.730
x to the fourth divided by
x to the third is just x.

00:03:25.730 --> 00:03:28.590
Times x, as x
approaches infinity.

00:03:28.590 --> 00:03:35.340
Or we could even say this
is going to be 4/250 times

00:03:35.340 --> 00:03:40.170
the limit, as x
approaches infinity of x.

00:03:40.170 --> 00:03:40.980
Now what's this?

00:03:40.980 --> 00:03:43.820
What's the limit of x as
x approaches infinity?

00:03:43.820 --> 00:03:45.810
Well, it's just going
to keep growing forever.

00:03:45.810 --> 00:03:47.290
So this is just going to
be, this right over here

00:03:47.290 --> 00:03:48.498
is just going to be infinity.

00:03:48.498 --> 00:03:50.460
Infinity times some
number right over here

00:03:50.460 --> 00:03:51.830
is going to be infinity.

00:03:51.830 --> 00:03:54.660
So the limit as x approaches
infinity of all of this,

00:03:54.660 --> 00:03:56.280
it's actually unbounded.

00:03:56.280 --> 00:03:57.541
It's infinity.

00:03:57.541 --> 00:03:59.540
And a kind of obvious way
of seeing that, right,

00:03:59.540 --> 00:04:03.800
from the get go, is to
realize that the numerator has

00:04:03.800 --> 00:04:04.921
a fourth degree term.

00:04:04.921 --> 00:04:06.920
While the highest degree
term in the denominator

00:04:06.920 --> 00:04:08.560
is only a third degree term.

00:04:08.560 --> 00:04:10.510
So the numerator is
going to grow far faster

00:04:10.510 --> 00:04:12.000
than the denominator.

00:04:12.000 --> 00:04:14.964
So if the numerator
is growing far faster

00:04:14.964 --> 00:04:16.380
than the denominator,
you're going

00:04:16.380 --> 00:04:18.769
to approach infinity
in this case.

00:04:18.769 --> 00:04:23.555
If the numerator is growing
slower than the denominator,

00:04:23.555 --> 00:04:26.530
if the denominator is growing
far faster than the numerator,

00:04:26.530 --> 00:04:29.790
like this case, you
are then approaching 0.

00:04:29.790 --> 00:04:32.750
So hopefully you find
that a little bit useful.


Limits by factoring

WEBVTT
Kind: captions
Language: en

00:00:00.620 --> 00:00:09.260
Let's say that f of x is equal
to x squared plus x minus 6

00:00:09.260 --> 00:00:12.470
over x minus 2.

00:00:12.470 --> 00:00:15.610
And we're curious about
what the limit of f

00:00:15.610 --> 00:00:21.540
of x, as x approaches
2, is equal to.

00:00:21.540 --> 00:00:23.450
Now the first attempt
that you might

00:00:23.450 --> 00:00:25.700
want to do right when you
see something like this, is

00:00:25.700 --> 00:00:28.040
just see what happens
what is f of 2.

00:00:28.040 --> 00:00:30.760
Now this won't always be the
limit, even if it's defined,

00:00:30.760 --> 00:00:32.259
but it's a good
place to start, just

00:00:32.259 --> 00:00:35.130
to see if it's something
reasonable could pop out.

00:00:35.130 --> 00:00:38.720
So looking at it this way,
if we just evaluate f of 2,

00:00:38.720 --> 00:00:42.570
on our numerator, we get
2 squared plus 2 minus 6.

00:00:42.570 --> 00:00:44.912
So it's going to be 4 plus
2, which is 6, minus 6,

00:00:44.912 --> 00:00:46.620
so you're going to
get 0 in the numerator

00:00:46.620 --> 00:00:49.310
and you're going to get
0 in the denominator.

00:00:49.310 --> 00:00:52.490
So we don't have, the
function is not defined,

00:00:52.490 --> 00:00:55.120
so not defined at x is equal 2.

00:00:55.120 --> 00:00:58.300
f not defined.

00:00:58.300 --> 00:00:59.810
So there's no
simple thing there.

00:00:59.810 --> 00:01:02.900
Even if this did evaluate, if
it was a continuous function,

00:01:02.900 --> 00:01:05.579
then the limit would be
whatever the function is,

00:01:05.579 --> 00:01:07.370
but that doesn't
necessarily mean the case.

00:01:07.370 --> 00:01:10.331
But we see very clearly the
function is not defined here.

00:01:10.331 --> 00:01:11.830
So let's see if we
can simplify this

00:01:11.830 --> 00:01:13.754
and also try to
graph it in some way.

00:01:13.754 --> 00:01:15.920
So one thing that might
have jumped out at your head

00:01:15.920 --> 00:01:18.906
is you might want to factor
this expression on top.

00:01:18.906 --> 00:01:20.280
So if we want to
rewrite this, we

00:01:20.280 --> 00:01:22.270
can rewrite the top expression.

00:01:22.270 --> 00:01:24.620
And this just goes back
to your algebra one,

00:01:24.620 --> 00:01:26.650
two numbers whose
product is negative 6,

00:01:26.650 --> 00:01:30.250
whose sum is positive 3,
well that could be positive 3

00:01:30.250 --> 00:01:31.100
and negative 2.

00:01:31.100 --> 00:01:35.450
So this could be
x plus 3 times x

00:01:35.450 --> 00:01:40.940
minus 2, all of
that over x minus 2.

00:01:40.940 --> 00:01:43.950
So as long as x
does not equal 2,

00:01:43.950 --> 00:01:46.460
these two things
will cancel out.

00:01:46.460 --> 00:01:53.140
So we could say this is equal
to x plus 3 for all X's except

00:01:53.140 --> 00:01:54.330
for x is equal to 2.

00:01:57.976 --> 00:01:59.600
So that's another
way of looking at it.

00:01:59.600 --> 00:02:02.182
Another way we could
rewrite our f of x,

00:02:02.182 --> 00:02:04.440
we'll do it in blue, just
to change the colors,

00:02:04.440 --> 00:02:07.230
we could rewrite f of x, this
is the exact same function,

00:02:07.230 --> 00:02:14.340
f of x is equal to x plus
3 when x does not equal 2.

00:02:14.340 --> 00:02:20.410
And we could even say it's
undefined when x is equal to 2.

00:02:20.410 --> 00:02:23.360
So given this definition,
it becomes much clearer

00:02:23.360 --> 00:02:27.080
to us of how we can
actually graph f of x.

00:02:27.080 --> 00:02:29.860
So let's try to do it.

00:02:29.860 --> 00:02:31.870
So that is, that is
not anywhere near being

00:02:31.870 --> 00:02:34.380
a straight line,
that is much better.

00:02:34.380 --> 00:02:38.510
So let's call this the y-axis
call it y equals f of x.

00:02:38.510 --> 00:02:40.430
And then let's,
over here, let me

00:02:40.430 --> 00:02:44.670
make a horizontal line,
that is my x-axis.

00:02:44.670 --> 00:02:48.610
So defined this way, f of
x is equal to x plus 3.

00:02:48.610 --> 00:02:53.870
So if this is 1, 2, 3, we
have a y-intercept at 3

00:02:53.870 --> 00:02:54.880
and then the slope is 1.

00:02:57.800 --> 00:03:00.770
And it's defined for all X's
except for x is equal to 2.

00:03:00.770 --> 00:03:03.900
So this is x is equal
to 1, x is equal to 2.

00:03:03.900 --> 00:03:07.394
So when x is equal
to 2 it is undefined.

00:03:07.394 --> 00:03:08.810
So let me make
sure I can, so it's

00:03:08.810 --> 00:03:10.420
undefined right over there.

00:03:17.210 --> 00:03:22.500
So this is what f
of x looks like.

00:03:22.500 --> 00:03:25.190
Now given this, let's try
to answer our question.

00:03:25.190 --> 00:03:27.965
What is the limit of f
of x as x approaches 2.

00:03:27.965 --> 00:03:30.330
Well, we can look
at this graphically.

00:03:30.330 --> 00:03:34.740
As x approaches 2 from
lower values in 2,

00:03:34.740 --> 00:03:38.220
so this right over here is x is
equal to 2, if we get to maybe,

00:03:38.220 --> 00:03:41.840
let's say this is 1.7,
we see that our f of x

00:03:41.840 --> 00:03:43.270
is right over there.

00:03:43.270 --> 00:03:46.810
If we get to 1.9, our f
of x is right over there.

00:03:46.810 --> 00:03:52.000
So it seems to be approaching
this value right over there.

00:03:52.000 --> 00:03:56.220
Similarly, as we approach 2
from values greater than it,

00:03:56.220 --> 00:03:59.960
if we're at, I don't know,
this could be like 2.5,

00:03:59.960 --> 00:04:02.140
2.5 our f of x is
right over there.

00:04:02.140 --> 00:04:07.000
If we get even closer to 2,
our f of x is right over there.

00:04:07.000 --> 00:04:09.650
And once again, we look like
we are approaching this value.

00:04:09.650 --> 00:04:11.150
Or another way of
thinking about it,

00:04:11.150 --> 00:04:13.690
if we ride this line from
the positive direction,

00:04:13.690 --> 00:04:15.820
we seem to be approaching
this value for f

00:04:15.820 --> 00:04:18.990
of x, if we write this line
from the negative direction,

00:04:18.990 --> 00:04:22.180
from values less than 2,
we seem to be approaching

00:04:22.180 --> 00:04:24.880
this value right over here.

00:04:24.880 --> 00:04:28.310
And this is essentially
the value of x plus 3

00:04:28.310 --> 00:04:30.440
if we set x is equal to 2.

00:04:30.440 --> 00:04:33.850
So this is essentially going to
be, this value right over here,

00:04:33.850 --> 00:04:35.920
is equal to 5.

00:04:35.920 --> 00:04:37.930
If we just look at it
visually, if we just

00:04:37.930 --> 00:04:43.330
graphed a line with slope 1
with the y-intercept of 3,

00:04:43.330 --> 00:04:45.650
this value right over here is 5.

00:04:45.650 --> 00:04:50.750
Now we could also try to
do this it numerically,

00:04:50.750 --> 00:04:52.350
so let's try to do that.

00:04:52.350 --> 00:04:55.240
So if this is our
function definition,

00:04:55.240 --> 00:05:00.330
completely identical to
our original definition,

00:05:00.330 --> 00:05:03.940
then we just try values as x
gets closer and closer to 2.

00:05:03.940 --> 00:05:05.850
So let's try values less than 2.

00:05:05.850 --> 00:05:08.980
So 1.9999, and this
is almost obvious.

00:05:08.980 --> 00:05:15.960
1.9999 plus 3, well, that gets
you pretty darn close to 5.

00:05:15.960 --> 00:05:18.230
If I put even more 9s
here, get even closer to 2,

00:05:18.230 --> 00:05:20.320
we'd get even closer to 5 here.

00:05:20.320 --> 00:05:23.590
If we approach 2 from
the positive direction,

00:05:23.590 --> 00:05:27.370
and then, we once again, we're
getting closer and closer

00:05:27.370 --> 00:05:29.160
to 5 from the
positive direction.

00:05:29.160 --> 00:05:32.249
If we were even closer to
2, we'd be even closer to 5.

00:05:32.249 --> 00:05:33.790
So whether we look
at it numerically,

00:05:33.790 --> 00:05:36.660
or we look at it graphically,
it looks pretty clear

00:05:36.660 --> 00:05:43.780
that the limit here is
going to be equal to five.


Limit at a point of discontinuity

WEBVTT
Kind: captions
Language: en

00:00:00.702 --> 00:00:08.503
Let's say that f(x) is equal to the absolute value of x minus three over x minus three

00:00:08.503 --> 00:00:18.469
and what I'm curious about is the limit of f(x) as x approaches three

00:00:18.469 --> 00:00:26.767
and just from an inspection you can see that the function is not defined when x is equal to three - you get zero over zero: it's not defined

00:00:26.767 --> 00:00:32.978
So to answer this question let's try to re-write the same exact function definition slightly differently

00:00:32.978 --> 00:00:37.768
So let's say f(x) is going to be equal to - and I'm going to think of two cases:

00:00:37.768 --> 00:00:40.035
I'm going to think of the case when x is greater than three

00:00:40.035 --> 00:00:42.700
and when x is less than three

00:00:42.700 --> 00:00:47.142
So when x is - I'll do this in two different colors actually

00:00:47.142 --> 00:00:51.367
When x - I'll do it in green - that's not green

00:00:51.367 --> 00:00:53.767
When x is greater than three...

00:00:53.767 --> 00:00:57.434
When x is greater than three, what does this function simplify to?

00:00:57.434 --> 00:00:59.835
Well, whatever I get up here, I'm just taking the..

00:00:59.835 --> 00:01:02.375
I'm going to get a positive value up here and then I'm...

00:01:02.375 --> 00:01:05.421
Well, if I take the absolute value it's going to be the exact same thing, so let me...

00:01:05.421 --> 00:01:13.412
For x is greater than three, this is going to be the exact same thing as x minus three over x minus three

00:01:13.412 --> 00:01:21.300
because if x is greater than three, the numerator's going to be positive, you take the absolute value of that, you're not going to change its value

00:01:21.300 --> 00:01:25.500
so you get this right over here or, if we were to re-write it...

00:01:25.500 --> 00:01:31.633
...if we were to re-write it, this is equal to, for x is greater than three, you're going to have f(x) is equal to one

00:01:31.633 --> 00:01:33.696
for x is greater than three

00:01:33.696 --> 00:01:38.012
Similarly, let's think about what happens when x is less than three

00:01:38.012 --> 00:01:44.224
When x is less than three, well, x minus three is going to be a negative number

00:01:44.224 --> 00:01:47.471
When you take the absolute value of that, you're essentially negating it

00:01:47.471 --> 00:01:53.939
so it's going to be the negative of x minus three over x minus three

00:01:53.939 --> 00:01:58.644
or if you were to simplify these two things, for any value as long as x doesn't equal three

00:01:58.644 --> 00:02:03.235
this part right over her simplifies to one, so you are left with a negative one

00:02:03.235 --> 00:02:06.738
negative one for x is less than three

00:02:06.738 --> 00:02:09.761
I encourage you, if you don't believe what I just said, try it out with some numbers

00:02:09.761 --> 00:02:10.834
Try out some numbers:

00:02:10.834 --> 00:02:15.302
3.1, 3.001, 3.5, 4, 7

00:02:15.302 --> 00:02:17.900
Any number greater than three, you're going to get one

00:02:17.900 --> 00:02:19.963
You're going to get the same thing divided by the same thing

00:02:19.963 --> 00:02:21.969
and try values for x less than three:

00:02:21.969 --> 00:02:24.963
you're going to get negative one no matter what you try

00:02:24.963 --> 00:02:28.635
So let's visualise this function now

00:02:28.635 --> 00:02:32.070
So, now you draw some axes...

00:02:32.070 --> 00:02:33.849
That's my x- axis

00:02:33.849 --> 00:02:35.769
and then this is my...

00:02:35.769 --> 00:02:40.500
This is my f(x) axis - y is equal to f(x)

00:02:40.500 --> 00:02:42.633
and what we care about is x is equal to three

00:02:42.633 --> 00:02:47.635
so x is equal to one, two, three, four, five

00:02:47.635 --> 00:02:48.872
and we could keep going...

00:02:48.872 --> 00:02:53.233
and let's say this is positive one, two, so that's y is equal to one

00:02:53.233 --> 00:02:56.233
this is y is equal to negative one and negative two

00:02:56.233 --> 00:02:57.548
and we can keep going...

00:02:57.548 --> 00:02:59.969
So this way that we have re-written the function

00:02:59.969 --> 00:03:02.040
is the exact same function as this

00:03:02.040 --> 00:03:03.567
we've just written [it] in a different way

00:03:03.567 --> 00:03:05.003
and so what we're saying is...

00:03:05.003 --> 00:03:05.634
is we're...

00:03:05.634 --> 00:03:07.161
Our function is undefined at three

00:03:07.161 --> 00:03:10.796
but if our x is greater than three, our function is equal to one

00:03:10.796 --> 00:03:14.501
so if our x is greater than three, our function is equal to one

00:03:14.501 --> 00:03:15.752
so it looks like...

00:03:15.752 --> 00:03:19.838
It looks like that, and it's undefined at three

00:03:19.838 --> 00:03:23.312
and if x is less than three our function is equal to negative one

00:03:23.312 --> 00:03:26.678
so it looks like - I'll be doing that same color

00:03:26.678 --> 00:03:28.212
It looks like this...

00:03:28.212 --> 00:03:30.169
It looks like...

00:03:30.169 --> 00:03:31.567
Looks like this...

00:03:31.567 --> 00:03:34.836
Once again, it's undefined at three

00:03:34.836 --> 00:03:36.498
So it looks like that

00:03:36.498 --> 00:03:38.303
So now let's try to answer our question:

00:03:38.303 --> 00:03:41.833
What is the limit as x approaches three?

00:03:41.833 --> 00:03:44.230
Well, let's think about the limit as x approaches three

00:03:44.230 --> 00:03:47.567
from the negative direction, from values less than three

00:03:47.567 --> 00:03:49.924
So let's think about first the limit...

00:03:49.924 --> 00:03:52.831
...the limit, as x approaches three...

00:03:52.831 --> 00:03:57.767
...as x approaches three, the limit of f(x)...

00:03:57.767 --> 00:04:00.512
...as x approaches three from the negative direction

00:04:00.512 --> 00:04:04.400
and all this notation here - I wrote this negative as a superscript right after the three - says

00:04:04.400 --> 00:04:06.567
Let's think about the limit as we're approaching...

00:04:06.567 --> 00:04:07.636
...let me make this clear...

00:04:07.636 --> 00:04:11.069
Let's think about the limit as we're approaching from the left

00:04:11.069 --> 00:04:13.033
So in this case, if we get closer...

00:04:13.033 --> 00:04:14.235
If we get...

00:04:14.235 --> 00:04:16.450
If we start with values lower than three

00:04:16.450 --> 00:04:18.620
as we get closer and closer and closer...

00:04:18.620 --> 00:04:22.034
So, say we start at zero, f(x) is equal to negative one

00:04:22.034 --> 00:04:23.634
We go to one, f(x) is equal to negative one

00:04:23.634 --> 00:04:25.767
We go to two, f(x) is equal to negative one

00:04:25.767 --> 00:04:30.769
If you go to 2.999999, f(x) is equal to negative one

00:04:30.769 --> 00:04:33.380
So it looks like it is approaching negative one if you approach..

00:04:33.380 --> 00:04:37.367
...if you approach from the left-hand side

00:04:37.367 --> 00:04:39.285
Now let's think about the limit...

00:04:39.285 --> 00:04:41.625
...the limit of f(x)...

00:04:41.625 --> 00:04:48.690
...the limit of f(x) as x approaches three from the positive direction, from values greater than three

00:04:48.690 --> 00:04:53.149
So here we see, when x is equal to five, f(x) is equal to one

00:04:53.149 --> 00:04:55.968
When x is equal to four, f(x) is equal to one

00:04:55.968 --> 00:05:01.368
When x is equal to 3.0000001, f(x) is equal to one

00:05:01.368 --> 00:05:03.305
So it seems to be approaching...

00:05:03.305 --> 00:05:06.033
It seems to be approaching positive one

00:05:06.033 --> 00:05:08.101
So now we have something strange

00:05:08.101 --> 00:05:11.367
We seem to be approaching a different value when we approach from the left

00:05:11.367 --> 00:05:14.094
than when we approach from the right

00:05:14.094 --> 00:05:19.347
and if we are approaching two different values then the limit does not exist

00:05:19.347 --> 00:05:27.951
So this limit right over here does not exist

00:05:27.951 --> 00:05:30.209
or another way of saying it:

00:05:30.209 --> 00:05:31.403
The limit...

00:05:31.403 --> 00:05:34.165
...the limit of...

00:05:34.165 --> 00:05:37.563
(Let me write this in a new color - I have a little idea here)

00:05:37.563 --> 00:05:47.968
...the limit of a function f(x) as x approaches some value c is equal to L if and only if...

00:05:47.968 --> 00:05:57.763
...if and only if the limit of f(x) as x approaches c from the negative direction is equal to the limit

00:05:57.763 --> 00:06:06.033
of f(x) as x approaches c from the positive direction which is equal to L

00:06:06.033 --> 00:06:07.354
This did not happen here -

00:06:07.354 --> 00:06:10.093
the limit when we approached the left was negative one,

00:06:10.093 --> 00:06:12.351
the limit when we approached from the right was positive one,

00:06:12.441 --> 00:06:15.401
So we did not get the same limits when we approached from either side

00:06:15.401 --> 00:06:19.401
So the limit does not exist in this case


Determining which limit statements are true

WEBVTT
Kind: captions
Language: en

00:00:00.840 --> 00:00:03.767
So we have a function, f of
x, graphed right over here.

00:00:03.767 --> 00:00:05.350
And then we have a
bunch of statements

00:00:05.350 --> 00:00:08.789
about the limit of f of x, as
x approaches different values.

00:00:08.789 --> 00:00:11.330
And what I want to do is figure
out which of these statements

00:00:11.330 --> 00:00:14.140
are true and which
of these are false.

00:00:14.140 --> 00:00:16.400
So let's look at
this first statement.

00:00:16.400 --> 00:00:21.350
Limit of f of x, as x approaches
1 from the positive direction,

00:00:21.350 --> 00:00:23.580
is equal to 0.

00:00:23.580 --> 00:00:27.220
So is this true or false?

00:00:27.220 --> 00:00:28.670
So let's look at it.

00:00:28.670 --> 00:00:30.860
So we're talking about
as x approaches 1

00:00:30.860 --> 00:00:35.290
from the positive direction,
so for values greater than 1.

00:00:35.290 --> 00:00:38.690
So as x approaches 1 from
the positive direction,

00:00:38.690 --> 00:00:39.530
what is f of x?

00:00:39.530 --> 00:00:42.620
Well, when x is, let's say 1
and 1/2, f of x is up here,

00:00:42.620 --> 00:00:47.730
as x gets closer and closer
to 1, f of x stays right at 1.

00:00:47.730 --> 00:00:51.480
So as x approaches 1 from
the positive direction,

00:00:51.480 --> 00:00:56.510
it looks like the limit of
f of x as x approaches 1

00:00:56.510 --> 00:00:59.030
from the positive
direction isn't 0.

00:00:59.030 --> 00:01:01.100
It looks like it is 1.

00:01:01.100 --> 00:01:04.346
So this is not true.

00:01:04.346 --> 00:01:05.970
This would be true
if instead of saying

00:01:05.970 --> 00:01:07.390
from the positive
direction, we said

00:01:07.390 --> 00:01:08.556
from the negative direction.

00:01:08.556 --> 00:01:12.870
From the negative direction,
the value of the function

00:01:12.870 --> 00:01:15.580
really does look like
it is approaching 0.

00:01:15.580 --> 00:01:18.430
For approaching 1 from the
negative direction, when

00:01:18.430 --> 00:01:21.370
x is right over
here, this is f of x.

00:01:21.370 --> 00:01:23.650
When x is right over
here, this is f of x.

00:01:23.650 --> 00:01:26.280
When x is right over
here, this is f of x.

00:01:26.280 --> 00:01:27.880
And we see that
the value of f of x

00:01:27.880 --> 00:01:29.484
seems to get closer
and closer to 0.

00:01:29.484 --> 00:01:30.900
So this would only
be true if they

00:01:30.900 --> 00:01:33.410
were approaching from
the negative direction.

00:01:33.410 --> 00:01:34.660
Next question.

00:01:34.660 --> 00:01:38.010
Limit of f of x, as x approaches
0 from the negative direction,

00:01:38.010 --> 00:01:40.634
is the same as limit of
f of x as x approaches 0

00:01:40.634 --> 00:01:41.800
from the positive direction.

00:01:41.800 --> 00:01:44.700
Is this statement true?

00:01:44.700 --> 00:01:45.420
Well, let's look.

00:01:45.420 --> 00:01:48.060
Our function, f of
x, as we approach 0

00:01:48.060 --> 00:01:49.530
from the negative
direction-- I'm

00:01:49.530 --> 00:01:51.720
using a new color--
as we approach 0

00:01:51.720 --> 00:01:56.350
from the negative direction,
so right over here,

00:01:56.350 --> 00:01:57.730
this is our value of f of x.

00:01:57.730 --> 00:02:00.580
Then as we get closer, this
is our value of f of x.

00:02:00.580 --> 00:02:03.400
As we get even closer, this
is our value of f of x.

00:02:03.400 --> 00:02:05.410
So it seems from the
negative direction

00:02:05.410 --> 00:02:07.560
like it is approaching
positive 1.

00:02:07.560 --> 00:02:12.050
From the positive direction,
when x is greater than 0,

00:02:12.050 --> 00:02:13.070
let's try it out.

00:02:13.070 --> 00:02:15.980
So if, say, x is 1/2,
this is our f of x.

00:02:15.980 --> 00:02:19.820
If x is, let's say,
1/4, this is our f of x.

00:02:19.820 --> 00:02:23.070
If x is just barely larger
than 0, this is our f of x.

00:02:23.070 --> 00:02:26.050
So it also seems
to be approaching

00:02:26.050 --> 00:02:29.120
f of x is equal to 1.

00:02:29.120 --> 00:02:30.240
So this looks true.

00:02:30.240 --> 00:02:33.320
They both seem to be
approaching the limit of 1.

00:02:33.320 --> 00:02:34.460
The limit here is 1.

00:02:34.460 --> 00:02:38.010
So this is absolutely true.

00:02:38.010 --> 00:02:39.860
Now let's look at
this statement.

00:02:39.860 --> 00:02:41.740
The limit of f of
x, as x approaches 0

00:02:41.740 --> 00:02:44.365
from the negative
direction, is equal to 1.

00:02:44.365 --> 00:02:45.990
Well, we've already
thought about that.

00:02:45.990 --> 00:02:52.080
The limit of f of
x, as x approaches 0

00:02:52.080 --> 00:02:53.760
from the negative
direction, we see

00:02:53.760 --> 00:02:55.570
that we're getting
closer and closer to 1.

00:02:55.570 --> 00:02:58.120
As x gets closer and
closer to 0, f of x

00:02:58.120 --> 00:02:59.470
gets closer and closer to 1.

00:02:59.470 --> 00:03:01.520
So this is also true.

00:03:01.520 --> 00:03:05.235
Limit of f of x, as x
approaches 0 exists.

00:03:05.235 --> 00:03:06.360
Well, it definitely exists.

00:03:06.360 --> 00:03:08.690
We've already established
that it's equal to 1.

00:03:08.690 --> 00:03:10.730
So that's true.

00:03:10.730 --> 00:03:13.880
Now the limit of f of x
as x approaches 1 exists,

00:03:13.880 --> 00:03:16.300
is that true?

00:03:16.300 --> 00:03:18.570
Well, we already saw that
when we were approaching 1

00:03:18.570 --> 00:03:20.380
from the positive
direction, the limit

00:03:20.380 --> 00:03:22.670
seems to be approaching 1.

00:03:22.670 --> 00:03:25.880
We get when x is 1
and 1/2, f of x is 1.

00:03:25.880 --> 00:03:28.031
When x is a little bit
more than 1, it's 1.

00:03:28.031 --> 00:03:30.280
So it seems like we're getting
closer and closer to 1.

00:03:30.280 --> 00:03:31.363
So let me write that down.

00:03:31.363 --> 00:03:36.140
The limit of f of
x, as x approaches 1

00:03:36.140 --> 00:03:38.680
from the positive
direction, is equal to 1.

00:03:38.680 --> 00:03:43.950
And now what's the limit
of f of x as x approaches 1

00:03:43.950 --> 00:03:45.910
from the negative direction?

00:03:45.910 --> 00:03:48.290
Well, here, this is our f of x.

00:03:48.290 --> 00:03:49.780
Here, this is our f of x.

00:03:49.780 --> 00:03:53.190
It seems like our f of x is
getting closer and closer to 0,

00:03:53.190 --> 00:03:56.050
when we approach 1 from
values less than 1.

00:03:56.050 --> 00:03:58.140
So over here it equals 0.

00:03:58.140 --> 00:04:00.670
So if the limit from
the right-hand side

00:04:00.670 --> 00:04:03.170
is a different value than the
limit from the left-hand side,

00:04:03.170 --> 00:04:05.330
then the limit does not exist.

00:04:05.330 --> 00:04:10.830
So this is not true.

00:04:10.830 --> 00:04:13.620
Now finally, the limit of f
of x, as x approaches 1.5,

00:04:13.620 --> 00:04:14.920
is equal to 1.

00:04:14.920 --> 00:04:16.015
So right over here.

00:04:16.015 --> 00:04:17.890
So everything we've been
dealing with so far,

00:04:17.890 --> 00:04:19.848
we've always looked at
points of discontinuity,

00:04:19.848 --> 00:04:22.930
or points where maybe the
function isn't quite defined.

00:04:22.930 --> 00:04:25.000
But here, this is kind
of a plain vanilla point.

00:04:25.000 --> 00:04:27.890
When x is equal to 1.5,
that's maybe right over here,

00:04:27.890 --> 00:04:30.052
this is f of 1.5.

00:04:30.052 --> 00:04:31.510
That right over
there is the point,

00:04:31.510 --> 00:04:33.380
well, this is the
value f of 1.5.

00:04:33.380 --> 00:04:39.770
We could say f of, we could see
that f of 1.5 is equal to 1,

00:04:39.770 --> 00:04:43.800
that this right here is
the point 1.5 comma 1.

00:04:43.800 --> 00:04:46.700
And if we approach it
from the left-hand side,

00:04:46.700 --> 00:04:50.071
from values less than it, it's
1, the limit seems to be 1.

00:04:50.071 --> 00:04:51.820
When we approach from
the right-hand side,

00:04:51.820 --> 00:04:52.850
the limit seems to be 1.

00:04:52.850 --> 00:04:54.830
So this is a pretty
straightforward thing.

00:04:54.830 --> 00:04:57.500
The graph is continuous
right there, and so really,

00:04:57.500 --> 00:04:59.210
if we just substitute
at that point,

00:04:59.210 --> 00:05:01.200
or we just look at
the graph, the limit

00:05:01.200 --> 00:05:02.870
is the value of
the function there.

00:05:02.870 --> 00:05:05.390
You don't have to have a
function be undefined in order

00:05:05.390 --> 00:05:06.730
to find a limit there.

00:05:06.730 --> 00:05:09.350
So it is, indeed, the
case that the limit

00:05:09.350 --> 00:05:15.630
of f of x, as x approaches
1.5, is equal to 1.


Limit properties

WEBVTT
Kind: captions
Language: en

00:00:00.339 --> 00:00:02.130
What I want to do in
this video is give you

00:00:02.130 --> 00:00:03.470
a bunch of properties of limits.

00:00:03.470 --> 00:00:05.470
And we're not going to
prove it rigorously here.

00:00:05.470 --> 00:00:07.420
In order to have the rigorous
proof of these properties,

00:00:07.420 --> 00:00:09.462
we need a rigorous definition
of what a limit is.

00:00:09.462 --> 00:00:11.211
And we're not doing
that in this tutorial,

00:00:11.211 --> 00:00:13.390
we'll do that in the
tutorial on the epsilon delta

00:00:13.390 --> 00:00:14.680
definition of limits.

00:00:14.680 --> 00:00:17.180
But most of these should
be fairly intuitive.

00:00:17.180 --> 00:00:19.840
And they are very helpful for
simplifying limit problems

00:00:19.840 --> 00:00:21.040
in the future.

00:00:21.040 --> 00:00:24.290
So let's say we know that
the limit of some function

00:00:24.290 --> 00:00:28.910
f of x, as x approaches
c, is equal to capital L.

00:00:28.910 --> 00:00:32.330
And let's say that we
also know that the limit

00:00:32.330 --> 00:00:36.580
of some other function, let's
say g of x, as x approaches c,

00:00:36.580 --> 00:00:39.270
is equal to capital M.

00:00:39.270 --> 00:00:43.170
Now given that, what
would be the limit of f

00:00:43.170 --> 00:00:49.820
of x plus g of x
as x approaches c?

00:00:49.820 --> 00:00:51.640
Well-- and you could
look at this visually,

00:00:51.640 --> 00:00:53.820
if you look at the graphs
of two arbitrary functions,

00:00:53.820 --> 00:00:55.560
you would essentially just
add those two functions--

00:00:55.560 --> 00:00:57.935
it'll be pretty clear that
this is going to be equal to--

00:00:57.935 --> 00:00:59.893
and once again, I'm not
doing a rigorous proof,

00:00:59.893 --> 00:01:01.960
I'm just really giving
you the properties here--

00:01:01.960 --> 00:01:06.760
this is going to be the limit
of f of x as x approaches c,

00:01:06.760 --> 00:01:11.630
plus the limit of g of
x as x approaches c.

00:01:11.630 --> 00:01:14.670
Which is equal to, well
this right over here

00:01:14.670 --> 00:01:17.880
is-- let me do that
in that same color--

00:01:17.880 --> 00:01:20.620
this right here is
just equal to L.

00:01:20.620 --> 00:01:26.190
It's going to be equal to L
plus M. This right over here

00:01:26.190 --> 00:01:27.530
is equal to M.

00:01:27.530 --> 00:01:28.890
Not too difficult.

00:01:28.890 --> 00:01:33.730
This is often called the sum
rule, or the sum property,

00:01:33.730 --> 00:01:35.167
of limits.

00:01:35.167 --> 00:01:37.750
And we could come up with a very
similar one with differences.

00:01:37.750 --> 00:01:44.220
The limit as x approaches
c of f of x minus g of x,

00:01:44.220 --> 00:01:47.550
is just going to be
L minus M. It's just

00:01:47.550 --> 00:01:49.330
the limit of f of
x as x approaches

00:01:49.330 --> 00:01:52.470
c, minus the limit of g
of x as x approaches c.

00:01:52.470 --> 00:01:58.594
So it's just going
to be L minus M.

00:01:58.594 --> 00:02:00.260
And we also often
call it the difference

00:02:00.260 --> 00:02:02.720
rule, or the difference
property, of limits.

00:02:02.720 --> 00:02:05.550
And these once again, are very,
very, hopefully, reasonably

00:02:05.550 --> 00:02:06.364
intuitive.

00:02:06.364 --> 00:02:08.780
Now what happens if you take
the product of the functions?

00:02:08.780 --> 00:02:16.990
The limit of f of x times
g of x as x approaches c.

00:02:16.990 --> 00:02:18.820
Well lucky for us,
this is going to be

00:02:18.820 --> 00:02:22.740
equal to the limit of
f of x as x approaches

00:02:22.740 --> 00:02:28.410
c, times the limit of g
of x, as x approaches c.

00:02:28.410 --> 00:02:31.110
Lucky for us, this is kind of
a fairly intuitive property

00:02:31.110 --> 00:02:32.022
of limits.

00:02:32.022 --> 00:02:33.480
So in this case,
this is just going

00:02:33.480 --> 00:02:41.030
to be equal to, this is
L times M. This is just

00:02:41.030 --> 00:02:45.040
going to be L times
M. Same thing,

00:02:45.040 --> 00:02:47.930
if instead of having a function
here, we had a constant.

00:02:47.930 --> 00:02:50.205
So if we just had
the limit-- let

00:02:50.205 --> 00:02:56.110
me do it in that same
color-- the limit of k times

00:02:56.110 --> 00:03:00.310
f of x, as x approaches c,
where k is just some constant.

00:03:00.310 --> 00:03:03.520
This is going to be the same
thing as k times the limit

00:03:03.520 --> 00:03:06.820
of f of x as x approaches c.

00:03:06.820 --> 00:03:13.960
And that is just equal
to L. So this whole thing

00:03:13.960 --> 00:03:20.070
simplifies to k times L.

00:03:20.070 --> 00:03:22.380
And we can do the same
thing with difference.

00:03:22.380 --> 00:03:25.520
This is often called the
constant multiple property.

00:03:25.520 --> 00:03:28.210
We can do the same
thing with differences.

00:03:28.210 --> 00:03:31.100
So if we have the
limit as x approaches

00:03:31.100 --> 00:03:35.200
c of f of x divided by g of x.

00:03:35.200 --> 00:03:37.900
This is the exact same
thing as the limit

00:03:37.900 --> 00:03:41.480
of f of x as x
approaches c, divided

00:03:41.480 --> 00:03:46.510
by the limit of g of
x as x approaches c.

00:03:46.510 --> 00:03:49.300
Which is going to be equal
to-- I think you get it now--

00:03:49.300 --> 00:03:56.750
this is going to be
equal to L over M.

00:03:56.750 --> 00:03:59.770
And finally-- this is sometimes
called the quotient property--

00:03:59.770 --> 00:04:01.982
finally we'll look at
the exponent property.

00:04:01.982 --> 00:04:07.090
So if I have the
limit of-- let me

00:04:07.090 --> 00:04:09.284
write it this way-- of
f of x to some power.

00:04:09.284 --> 00:04:10.700
And actually, let
me even write it

00:04:10.700 --> 00:04:12.880
as a fractional
power, to the r over s

00:04:12.880 --> 00:04:16.810
power, where both r and s are
integers, then the limit of f

00:04:16.810 --> 00:04:23.580
of x to the r over s
power as x approaches c,

00:04:23.580 --> 00:04:28.650
is going to be the exact
same thing as the limit of f

00:04:28.650 --> 00:04:33.670
of x as x approaches c
raised to the r over s power.

00:04:33.670 --> 00:04:36.620
Once again, when r and s
are both integers, and s

00:04:36.620 --> 00:04:37.670
is not equal to 0.

00:04:37.670 --> 00:04:40.350
Otherwise this exponent
would not make much sense.

00:04:40.350 --> 00:04:46.905
And this is the same thing
as L to the r over s power.

00:04:46.905 --> 00:04:54.850
So this is equal to L
to the r over s power.

00:04:54.850 --> 00:04:56.479
So using these, we
can actually find

00:04:56.479 --> 00:04:58.020
the limit of many,
many, many things.

00:04:58.020 --> 00:05:01.060
And what's neat about it is
the property of limits kind of

00:05:01.060 --> 00:05:03.360
are the things that you
would naturally want to do.

00:05:03.360 --> 00:05:05.068
And if you graph some
of these functions,

00:05:05.068 --> 00:05:07.450
it actually turns out
to be quite intuitive.


Visualizing derivatives exercise

WEBVTT
Kind: captions
Language: en

00:00:00.500 --> 00:00:04.130
The function f of x
is shown in green.

00:00:04.130 --> 00:00:06.840
The sliding purple window
may contain a section

00:00:06.840 --> 00:00:10.900
of an antiderivative of
the function, F of x.

00:00:10.900 --> 00:00:16.760
So, essentially it's
saying, this green function,

00:00:16.760 --> 00:00:19.110
or part of this green
function, is potentially

00:00:19.110 --> 00:00:22.080
the derivative of
this purple function.

00:00:22.080 --> 00:00:23.960
And what we need
to do is-- it says,

00:00:23.960 --> 00:00:25.835
where does the function
in the sliding window

00:00:25.835 --> 00:00:28.730
correspond to the
antiderivative of our function?

00:00:28.730 --> 00:00:32.729
The antiderivative of f of x,
usually, write as big F of x.

00:00:32.729 --> 00:00:36.980
This is just saying
that, lowercase f of x

00:00:36.980 --> 00:00:39.360
is just the derivative
of big F of x.

00:00:39.360 --> 00:00:42.777
So, at what point
could the derivative

00:00:42.777 --> 00:00:44.360
of the purple
function-- and I'm going

00:00:44.360 --> 00:00:46.710
to move the purple
function around-- where

00:00:46.710 --> 00:00:50.150
can the derivative of
that be the green stuff.

00:00:50.150 --> 00:00:52.752
So let's just focus on
the purple stuff first.

00:00:52.752 --> 00:00:54.376
So the derivative--
we can just view it

00:00:54.376 --> 00:00:56.990
as the slope of the tangent
line-- between this point

00:00:56.990 --> 00:01:00.130
and this point, we see that we
have a constant negative slope,

00:01:00.130 --> 00:01:03.370
and then we have a
constant positive slope.

00:01:03.370 --> 00:01:06.490
So let's see, where here do we
have a constant negative slope?

00:01:06.490 --> 00:01:09.430
Well, now here the slope is
0, and it gets more negative.

00:01:09.430 --> 00:01:11.450
Here we have a constant
positive slope, not

00:01:11.450 --> 00:01:13.006
a constant negative slope.

00:01:13.006 --> 00:01:14.630
Here we have a constant
negative slope,

00:01:14.630 --> 00:01:16.800
so maybe it matches
up over there.

00:01:16.800 --> 00:01:21.480
So here we have a
constant negative slope,

00:01:21.480 --> 00:01:25.660
but then on the purple function,
we have a positive slope,

00:01:25.660 --> 00:01:28.710
but where the potential
derivative is here,

00:01:28.710 --> 00:01:30.220
we just have a slope of 0.

00:01:30.220 --> 00:01:31.940
So, this doesn't
match up either.

00:01:31.940 --> 00:01:34.440
So it looks like in this case,
there's actually no solution.

00:01:34.440 --> 00:01:36.200
Let's see if this works out.

00:01:36.200 --> 00:01:36.990
Yes, correct.

00:01:36.990 --> 00:01:38.580
Next question.

00:01:38.580 --> 00:01:40.190
Let's do another one.

00:01:40.190 --> 00:01:42.470
A function f of x
is shown purple.

00:01:42.470 --> 00:01:44.020
The sliding green
window may contain

00:01:44.020 --> 00:01:45.750
a section of its derivative.

00:01:45.750 --> 00:01:47.942
So now we're trying
to say, at what

00:01:47.942 --> 00:01:49.400
point of this purple
function might

00:01:49.400 --> 00:01:52.530
the derivative look like
this green function?

00:01:52.530 --> 00:01:54.340
So in this green
function, if this

00:01:54.340 --> 00:01:56.540
is the function's
derivative, here the slope

00:01:56.540 --> 00:01:58.200
is very negative.

00:01:58.200 --> 00:02:02.180
It goes to 0, and then
the slope gets positive.

00:02:02.180 --> 00:02:03.520
So let's think about it.

00:02:03.520 --> 00:02:07.420
So over here, the slope is
just a constant negative,

00:02:07.420 --> 00:02:08.830
so that won't work.

00:02:08.830 --> 00:02:12.150
If we shift it over
here, our slope

00:02:12.150 --> 00:02:14.440
is very steep in the
negative direction

00:02:14.440 --> 00:02:15.940
and then it gets
less and less steep

00:02:15.940 --> 00:02:18.065
in the negative direction,
and it goes all the way,

00:02:18.065 --> 00:02:19.912
and then over here
the slope is 0.

00:02:19.912 --> 00:02:21.620
And over here, if this
is the derivative,

00:02:21.620 --> 00:02:23.811
it seems to match
up, the slope is 0.

00:02:23.811 --> 00:02:25.310
And then it gets
more and more steep

00:02:25.310 --> 00:02:26.410
in the positive direction.

00:02:26.410 --> 00:02:27.340
So this matches up.

00:02:27.340 --> 00:02:29.640
It looks like over
this interval,

00:02:29.640 --> 00:02:33.060
the green the function
is indeed the derivative

00:02:33.060 --> 00:02:35.350
of this purple function.

00:02:35.350 --> 00:02:35.930
So let's see.

00:02:35.930 --> 00:02:37.700
Let's check our answer.

00:02:37.700 --> 00:02:38.250
Correct.

00:02:38.250 --> 00:02:39.010
Next question.

00:02:39.010 --> 00:02:39.885
Let's do another one.

00:02:39.885 --> 00:02:41.430
This is exciting.

00:02:41.430 --> 00:02:43.222
A function f of x
is shown in green.

00:02:43.222 --> 00:02:45.180
The sliding purple window
may contain a section

00:02:45.180 --> 00:02:48.690
of an antiderivative of
the function, F of x.

00:02:48.690 --> 00:02:54.580
So, now we say, let's match
up this little purple section

00:02:54.580 --> 00:02:56.460
to its derivative.

00:02:56.460 --> 00:02:58.587
So the green is the
derivative, the purple function

00:02:58.587 --> 00:03:00.420
is the thing we're
taking the derivative of.

00:03:00.420 --> 00:03:01.795
So if we just look
at the purple,

00:03:01.795 --> 00:03:04.290
we see that we have a
constant negative slope

00:03:04.290 --> 00:03:07.324
in the first part of
it, then our slope-- so

00:03:07.324 --> 00:03:09.990
let me just look for where I can
find a constant negative slope.

00:03:09.990 --> 00:03:12.370
So here, this is a
constant positive slope.

00:03:12.370 --> 00:03:14.380
This is not a constant slope.

00:03:14.380 --> 00:03:16.120
This is a constant positive.

00:03:16.120 --> 00:03:17.860
Here's a constant
negative slope.

00:03:17.860 --> 00:03:19.230
Let's see if this works.

00:03:19.230 --> 00:03:22.200
So over this interval, between
here and here, my slope

00:03:22.200 --> 00:03:25.270
is a constant negative,
and indeed, it

00:03:25.270 --> 00:03:26.660
looks like a constant negative.

00:03:26.660 --> 00:03:28.680
And you see it's a
constant negative 1.

00:03:28.680 --> 00:03:30.960
And over here, you
see the derivative

00:03:30.960 --> 00:03:33.070
is right at negative
1, and it's constant,

00:03:33.070 --> 00:03:34.537
so that part looks good.

00:03:34.537 --> 00:03:36.370
And then when I look
at the purple function,

00:03:36.370 --> 00:03:38.610
my slope is 0
starting off, then it

00:03:38.610 --> 00:03:41.470
gets more and more steep
in the negative direction.

00:03:41.470 --> 00:03:44.800
And so my slope is 0, and it
gets more and more negative,

00:03:44.800 --> 00:03:48.440
so this is indeed
seems to match up.

00:03:48.440 --> 00:03:51.710
So, let's check our answer.

00:03:51.710 --> 00:03:52.740
Yes, got it right.

00:03:52.740 --> 00:03:53.740
I could keep doing this.

00:03:53.740 --> 00:03:55.890
This is so much fun.


Tax brackets and progressive taxation

WEBVTT
Kind: captions
Language: en

00:00:00.611 --> 00:00:03.201
Voiceover: You'll hear
the word "tax bracket"

00:00:03.201 --> 00:00:04.207
thrown around a lot,

00:00:04.207 --> 00:00:06.538
and there's often a misunderstanding

00:00:06.538 --> 00:00:09.193
of what a tax bracket is,

00:00:09.193 --> 00:00:11.140
so that's what we're
going to do in this video.

00:00:11.140 --> 00:00:13.276
I'm going to do it in a
very general, simple way

00:00:13.276 --> 00:00:16.433
that doesn't use the exact same tax code

00:00:16.433 --> 00:00:17.803
of any particular country,

00:00:17.803 --> 00:00:19.472
but most countries use

00:00:19.472 --> 00:00:21.739
some kind of a progressive tax system,

00:00:21.739 --> 00:00:24.701
that has some type of tax brackets.

00:00:24.701 --> 00:00:28.612
Let's say my country has a tax code

00:00:28.612 --> 00:00:36.789
that says between $0 of
income and $10,000 of income

00:00:36.789 --> 00:00:39.570
you pay no taxes on it,

00:00:39.570 --> 00:00:44.924
and then between 10,000
and, let say, 50,000,

00:00:44.924 --> 00:00:46.527
let me write that a little bit neater,

00:00:46.527 --> 00:00:49.859
between 10,000 and 50,000,

00:00:49.859 --> 00:00:52.854
so that's 50,000 right over there,

00:00:52.854 --> 00:00:56.010
you pay 20% of your income,

00:00:56.010 --> 00:00:56.860
and then let's just say,

00:00:56.860 --> 00:00:57.863
this is very simple;

00:00:57.863 --> 00:01:00.853
much simpler than any
actual tax codes out there;

00:01:00.853 --> 00:01:07.257
let's say anything above
$50,000, you pay 30%.

00:01:07.257 --> 00:01:09.322
30% of your income.

00:01:09.322 --> 00:01:11.457
The reason why I say that
there is confusion here

00:01:11.457 --> 00:01:12.458
is many people feel

00:01:12.458 --> 00:01:14.661
that if their income
throws them into a bracket,

00:01:14.661 --> 00:01:17.459
that they pay 20% on all of their money,

00:01:17.459 --> 00:01:18.864
but that's not the way it works.

00:01:18.864 --> 00:01:20.989
Let's just do a couple of examples.

00:01:20.989 --> 00:01:22.790
Let's say that someone earns ...

00:01:22.790 --> 00:01:24.524
let's say income,

00:01:25.862 --> 00:01:31.001
and then let's think about
how much taxes they will pay.

00:01:31.001 --> 00:01:34.523
Let's say someone makes $5,000 in income.

00:01:34.523 --> 00:01:36.257
How much taxes would they pay?

00:01:36.257 --> 00:01:38.727
The first $10,000 you pay no taxes,

00:01:38.727 --> 00:01:42.487
so they'd pay $0 in taxes.

00:01:42.487 --> 00:01:45.657
Let's say someone has $15,000.

00:01:45.657 --> 00:01:48.051
$15,000 in income.

00:01:48.051 --> 00:01:50.389
How much taxes would they pay?

00:01:50.389 --> 00:01:54.991
They would pay 0% on the first $10,000,

00:01:54.991 --> 00:01:59.789
so it would be 0% x $10,000.

00:01:59.789 --> 00:02:02.321
x £10,000,

00:02:02.321 --> 00:02:03.909
and then the next 5,000,

00:02:03.909 --> 00:02:05.896
they're going to be right over here,

00:02:05.896 --> 00:02:08.923
the next 5,000 they're going to pay 20%.

00:02:08.923 --> 00:02:15.401
So, + 20% x the next $5,000.

00:02:15.401 --> 00:02:17.354
The next $5,000.

00:02:17.354 --> 00:02:18.723
They'll pay, this is just 0,

00:02:18.723 --> 00:02:20.928
then 20% of $5,000.

00:02:20.928 --> 00:02:24.666
They will pay $1,000 in taxes.

00:02:24.666 --> 00:02:26.384
The misconception that many people have

00:02:26.384 --> 00:02:29.669
is, "Oh, $15,000 puts me
into this middle bracket,

00:02:29.669 --> 00:02:34.080
"so I'm going to have to
pay 20% on all $15,000,"

00:02:34.080 --> 00:02:35.856
and that's not the way it works.

00:02:35.856 --> 00:02:38.300
You only pay 20% for the increment

00:02:38.300 --> 00:02:39.880
that throws you in that bracket.

00:02:39.880 --> 00:02:42.586
Just for the $5,000.

00:02:42.586 --> 00:02:44.547
Let's do another example.

00:02:44.547 --> 00:02:47.993
Let's say that someone makes $100,000.

00:02:47.993 --> 00:02:51.326
Let's say that someone makes $100,000.

00:02:51.326 --> 00:02:53.059
$100,000.

00:02:53.059 --> 00:02:54.913
How much taxes would they pay?

00:02:54.913 --> 00:02:57.281
It wouldn't be that they're
in this last bracket,

00:02:57.281 --> 00:03:01.029
so they'd pay 30% on
their entire $100,000.

00:03:01.029 --> 00:03:03.861
They pay 0% on the first 10,000.

00:03:03.861 --> 00:03:08.281
0% x 10; I'll just write "10K" for 10,000.

00:03:08.281 --> 00:03:12.529
Then they'll pay 20% on the next 40,000.

00:03:12.529 --> 00:03:14.257
This range right over here.

00:03:14.257 --> 00:03:19.255
So, + 20% x the 40,000,

00:03:19.255 --> 00:03:22.653
and then they only pay 30% on
the increment of their income

00:03:22.653 --> 00:03:26.086
that is above this $50,000 threshold.

00:03:26.086 --> 00:03:30.665
Then they will pay 30%.

00:03:30.665 --> 00:03:34.665
They will pay 30% x the next $50,000,

00:03:34.665 --> 00:03:36.259
so 10,000 + 40 + 50.

00:03:36.259 --> 00:03:38.862
That's what adds up to $100,000.

00:03:38.862 --> 00:03:40.454
x $50,000.

00:03:40.454 --> 00:03:42.586
This adds up to, well, this is nothing.

00:03:42.586 --> 00:03:49.082
20% of $40,000 is going to be $8,000.

00:03:49.082 --> 00:03:57.088
Then 30% of $50,000 is going to be 15,000.

00:03:57.088 --> 00:03:59.515
Is going to be $15,000.

00:03:59.515 --> 00:04:01.835
In this situation, the
person will not pay 30%

00:04:01.835 --> 00:04:03.526
on the entire 100,000.

00:04:03.526 --> 00:04:07.010
They will pay 8,000 + $15,000,

00:04:07.010 --> 00:04:14.462
which is equal to $23,000 in income taxes.