The US Constitution

WEBVTT
Kind: captions
Language: en

00:00:00.000 --> 00:00:02.035
- [Teacher] In the last
video we discussed the great

00:00:02.035 --> 00:00:06.066
compromise made at the
Constitutional Convention in 1787,

00:00:06.066 --> 00:00:09.927
where delegates who were
trying to craft a new

00:00:09.927 --> 00:00:14.128
governmental system for the
United States agreed on how

00:00:14.128 --> 00:00:18.198
the legislative branch of the
government would be set up,

00:00:18.198 --> 00:00:22.398
and the agreement they made was
that it would be a two house

00:00:22.398 --> 00:00:26.565
or bicameral legislature with
a House of Representatives,

00:00:29.291 --> 00:00:32.060
where the representatives
would be apportioned

00:00:32.060 --> 00:00:36.093
based on state population,
and then a Senate where

00:00:36.093 --> 00:00:40.260
every state would get two
senators regardless of its size.

00:00:41.658 --> 00:00:44.495
This is just one example
of how the framers

00:00:44.495 --> 00:00:48.718
of the constitution tried to
introduce a notion of balance.

00:00:48.718 --> 00:00:51.856
In this video, I wanna zoom
out a little bit and look at

00:00:51.856 --> 00:00:55.873
the broader constitution,
because the legislative branch

00:00:55.873 --> 00:00:57.973
was really only one part of it.

00:00:57.973 --> 00:01:00.988
In fact, there are seven more
articles of the constitution.

00:01:00.988 --> 00:01:04.359
So here, I'd like to spend
some time taking a closer look

00:01:04.359 --> 00:01:07.706
at some of the other articles,
paying special attention

00:01:07.706 --> 00:01:11.592
to the executive branch
and the judicial branch,

00:01:11.592 --> 00:01:14.507
but before we do that, I
just want to take a moment

00:01:14.507 --> 00:01:17.887
to marvel at the size of the constitution.

00:01:17.887 --> 00:01:22.054
Not because it's so big,
but because it's so small.

00:01:23.848 --> 00:01:26.517
This is the first page
of the constitution,

00:01:26.517 --> 00:01:29.715
famously starting with, "We
the people," but the entire

00:01:29.715 --> 00:01:32.995
original constitution
could fit on four pages.

00:01:32.995 --> 00:01:37.099
Compare that to the constitutions
of many other nations

00:01:37.099 --> 00:01:39.878
which are hundreds of pages long,

00:01:39.878 --> 00:01:43.729
and I think the idea here
in having a constitution

00:01:43.729 --> 00:01:46.873
that's really only seven articles long

00:01:46.873 --> 00:01:49.804
was that it was gonna set down principles.

00:01:49.804 --> 00:01:54.075
This wasn't going to be a
whole set of laws that outlined

00:01:54.075 --> 00:01:58.262
everything that a state
should do in any situation

00:01:58.262 --> 00:02:01.847
but rather a set of
broad ideas around which

00:02:01.847 --> 00:02:04.597
lawmaking decisions could happen.

00:02:07.005 --> 00:02:09.112
In a way, you could
think of the constitution

00:02:09.112 --> 00:02:12.527
as being kind of broad
enough to be flexible.

00:02:12.527 --> 00:02:16.378
They spoke in larger
generalities that could be

00:02:16.378 --> 00:02:19.409
applied to many different situations,

00:02:19.409 --> 00:02:23.576
and I think the proof that
this was a good way to think

00:02:25.451 --> 00:02:28.467
about putting together a
constitution is just in the fact

00:02:28.467 --> 00:02:31.292
that we'd still have
this constitution today,

00:02:31.292 --> 00:02:35.338
more than 200 years after
it was written in 1787.

00:02:35.338 --> 00:02:39.481
The US Constitution is the
oldest constitution in the world

00:02:39.481 --> 00:02:43.314
that is still in effect
at the national level.

00:02:44.834 --> 00:02:47.584
I think that's a pretty big deal.

00:02:48.594 --> 00:02:50.825
How did this constitution work?

00:02:50.825 --> 00:02:53.900
Well, let's look a little bit
more closely at these first

00:02:53.900 --> 00:02:57.954
three articles and the branches
of government they created.

00:02:57.954 --> 00:03:01.520
One of the ways that the
framers of the constitution

00:03:01.520 --> 00:03:05.687
attempted to remedy the problems
caused by the single branch

00:03:07.050 --> 00:03:09.665
government under the
Articles of Confederation

00:03:09.665 --> 00:03:12.612
was creating a three branch government.

00:03:12.612 --> 00:03:15.373
One branch, established in Article 1,

00:03:15.373 --> 00:03:18.126
would be the Congress
and within this building

00:03:18.126 --> 00:03:22.030
is the House of
Representatives and the Senate.

00:03:22.030 --> 00:03:25.751
And this would be in
the eyes of the framers,

00:03:25.751 --> 00:03:29.218
really the most powerful of the branches.

00:03:29.218 --> 00:03:32.607
They gave Congress the power to make law,

00:03:32.607 --> 00:03:35.940
to tax, to raise an army, to coin money.

00:03:43.185 --> 00:03:45.499
They really envisioned
that most of the day to day

00:03:45.499 --> 00:03:49.337
operations and most of
the power of government

00:03:49.337 --> 00:03:52.670
would fall under the duties of Congress,

00:03:53.653 --> 00:03:57.201
but one thing that the
Articles of Confederation

00:03:57.201 --> 00:04:00.140
had lacked was a powerful executive,

00:04:00.140 --> 00:04:02.802
so the second branch of government,

00:04:02.802 --> 00:04:05.938
established in Article 2
is the executive branch,

00:04:05.938 --> 00:04:08.369
the head of which is the president.

00:04:08.369 --> 00:04:11.906
The job of the executive
would be to enforce

00:04:11.906 --> 00:04:15.156
or carry out the laws made by Congress,

00:04:16.281 --> 00:04:20.434
and that would include doing
things like waging war.

00:04:20.434 --> 00:04:23.714
Remember that the first
president was George Washington,

00:04:23.714 --> 00:04:27.521
who had been the General of
the Revolutionary Armies,

00:04:27.521 --> 00:04:30.900
but the president could also
kind of have the front lines

00:04:30.900 --> 00:04:35.067
on dealing with foreign nations,
so negotiating treaties,

00:04:37.146 --> 00:04:39.429
and would also have the power

00:04:39.429 --> 00:04:42.762
of appointing many government officials.

00:04:45.594 --> 00:04:48.936
And lastly, the third branch
would be the judicial branch

00:04:48.936 --> 00:04:51.718
of government, established in Article 3.

00:04:51.718 --> 00:04:55.199
The head of the judicial branch
would be the Supreme Court,

00:04:55.199 --> 00:04:57.114
the highest court in the land.

00:04:57.114 --> 00:04:59.424
Of course, there are
many other smaller courts

00:04:59.424 --> 00:05:02.298
below it at the state and district level.

00:05:02.298 --> 00:05:05.063
The Supreme Court's job,
along with other courts,

00:05:05.063 --> 00:05:09.058
would be to interpret the
law, to see whether or not

00:05:09.058 --> 00:05:12.086
things done by Congress and the president

00:05:12.086 --> 00:05:15.289
fell within the bounds
of the constitution.

00:05:15.289 --> 00:05:19.178
And the framers really thought
that the judicial branch

00:05:19.178 --> 00:05:21.992
would end up being the
weakest branch of government,

00:05:21.992 --> 00:05:25.977
although both the presidency
and the Supreme Court

00:05:25.977 --> 00:05:28.546
have grown in power over the years.

00:05:28.546 --> 00:05:31.589
Now, this is an incredibly brief overview

00:05:31.589 --> 00:05:33.314
of these three branches.

00:05:33.314 --> 00:05:36.298
These articles include lots more in them

00:05:36.298 --> 00:05:39.597
about the specific powers
of each of these branches

00:05:39.597 --> 00:05:42.912
and the kinds of
requirements one would need

00:05:42.912 --> 00:05:45.612
to become a representative or president,

00:05:45.612 --> 00:05:48.457
so I highly recommend
that you read more about

00:05:48.457 --> 00:05:51.045
the Constitution and
check out these articles.

00:05:51.045 --> 00:05:53.768
But what I want you to get out
of this is that the framers

00:05:53.768 --> 00:05:57.909
here were trying to separate
the powers of government,

00:05:57.909 --> 00:06:01.236
so they wanted to make sure that to avoid

00:06:01.236 --> 00:06:04.263
having too much power in government.

00:06:04.263 --> 00:06:08.229
Remember that they are trying
to escape from the monarchy.

00:06:08.229 --> 00:06:11.964
They want to make sure
that government power

00:06:11.964 --> 00:06:15.761
is kind of diffused among
these three branches

00:06:15.761 --> 00:06:19.007
with the idea that they're
going to have to argue

00:06:19.007 --> 00:06:21.978
with each other to get things
done, they're going to have to

00:06:21.978 --> 00:06:25.007
cooperate with each
other to get things done,

00:06:25.007 --> 00:06:28.800
so the separation of
powers is one of the key

00:06:28.800 --> 00:06:31.176
principles of the Constitution.

00:06:31.176 --> 00:06:34.792
Another key principle
is checks and balances.

00:06:34.792 --> 00:06:37.816
What do I mean by checks and balances?

00:06:37.816 --> 00:06:41.807
Well, this is the idea
that each of the branches

00:06:41.807 --> 00:06:45.974
of government has the power
to check in the sense of stop,

00:06:48.461 --> 00:06:52.444
like checkmate in chess, the
other branches of government.

00:06:52.444 --> 00:06:55.658
I think of this as kind of like a giant

00:06:55.658 --> 00:06:59.241
governmental game of
rock, paper, scissors.

00:07:01.176 --> 00:07:04.880
Now, there are many ways
that these branches can

00:07:04.880 --> 00:07:07.711
check each other, but I just
wanna give a couple of quick

00:07:07.711 --> 00:07:10.843
examples to help you understand
what that might be like.

00:07:10.843 --> 00:07:14.093
Alright, say that Congress makes a law,

00:07:15.752 --> 00:07:19.020
and the president doesn't like that law.

00:07:19.020 --> 00:07:21.298
The president can use the power

00:07:21.298 --> 00:07:23.715
of the veto to kill that law,

00:07:25.273 --> 00:07:28.742
and if Congress gets annoyed
enough with the president,

00:07:28.742 --> 00:07:32.908
they might use their power
to impeach the president.

00:07:32.908 --> 00:07:35.255
What about the judicial branch?

00:07:35.255 --> 00:07:38.935
The judicial branch's
main checking function

00:07:38.935 --> 00:07:41.852
is declaring laws unconstitutional.

00:07:45.142 --> 00:07:48.108
The president or Congress may put through

00:07:48.108 --> 00:07:50.857
a law that the Supreme Court says

00:07:50.857 --> 00:07:54.190
is not consistent with the Constitution.

00:07:56.051 --> 00:07:58.556
The judicial branch can then kill that law

00:07:58.556 --> 00:08:01.065
by declaring it unconstitutional.

00:08:01.065 --> 00:08:03.055
What happens if the other
branches are unhappy

00:08:03.055 --> 00:08:04.933
with the judicial branch?

00:08:04.933 --> 00:08:07.522
One way that the president
can check the judicial branch

00:08:07.522 --> 00:08:09.863
is through appointing judges.

00:08:09.863 --> 00:08:14.030
This would kind of change
the composition of the court,

00:08:16.459 --> 00:08:19.559
the people on the court, and so over time,

00:08:19.559 --> 00:08:23.787
the presidency can influence
who is on the Supreme Court

00:08:23.787 --> 00:08:25.967
and how they rule on laws.

00:08:25.967 --> 00:08:30.006
And lastly, if Congress isn't
happy with the Supreme Court,

00:08:30.006 --> 00:08:33.235
they might be able to impeach justices,

00:08:33.235 --> 00:08:36.818
or change jurisdiction
of the lower courts.

00:08:39.543 --> 00:08:42.533
In this way, like the
separation of powers,

00:08:42.533 --> 00:08:45.644
the framers intended to make sure that

00:08:45.644 --> 00:08:48.364
one branch couldn't get too powerful,

00:08:48.364 --> 00:08:52.448
because it would be able to
check the other branches.

00:08:52.448 --> 00:08:56.615
You could really think of
this as being a brilliant way

00:08:58.568 --> 00:09:02.089
of harnessing peoples' natural inclination

00:09:02.089 --> 00:09:04.384
to look out for themselves.

00:09:04.384 --> 00:09:08.144
As rivalries developed,
as people tried to do

00:09:08.144 --> 00:09:11.991
what they thought was
best, they could help keep

00:09:11.991 --> 00:09:15.058
government honest by
fighting among themselves,

00:09:15.058 --> 00:09:18.116
and as they all strove
and checked each other,

00:09:18.116 --> 00:09:22.283
it would keep all of government
from becoming too powerful.


The Constitutional Convention

WEBVTT
Kind: captions
Language: en

00:00:00.218 --> 00:00:01.700
- [Instructor] In the United States today,

00:00:01.700 --> 00:00:05.083
we know our system of government so well

00:00:05.083 --> 00:00:07.175
that it hardly bears thinking about.

00:00:07.175 --> 00:00:09.482
We know that there's a president

00:00:09.482 --> 00:00:12.274
who's the head of the Executive Branch,

00:00:12.274 --> 00:00:14.050
there's Congress, which is made up

00:00:14.050 --> 00:00:16.424
of the House of
Representatives and the Senate,

00:00:16.424 --> 00:00:20.092
and there's the judicial branch,
which has the Supreme Court

00:00:20.092 --> 00:00:23.016
as its head of a whole court system

00:00:23.016 --> 00:00:25.242
that stretches throughout
the United States.

00:00:25.242 --> 00:00:29.299
But how did the United States
end up with this system?

00:00:29.299 --> 00:00:33.483
We frequently forget that
the Constitutional Convention

00:00:33.483 --> 00:00:37.650
which created this system we
know today happened in 1787.

00:00:38.772 --> 00:00:41.022
That was more than a decade

00:00:42.017 --> 00:00:44.930
after the Declaration of Independence.

00:00:44.930 --> 00:00:48.524
So there was this 11-year plus period

00:00:48.524 --> 00:00:52.233
before the United States had
its modern day constitution

00:00:52.233 --> 00:00:56.957
and during that time, fought
the Revolutionary War,

00:00:56.957 --> 00:00:57.957
won in 1783,

00:00:59.581 --> 00:01:02.725
and tried out a completely
different system of government

00:01:02.725 --> 00:01:04.875
called the Articles of Confederation,

00:01:04.875 --> 00:01:08.557
which we talked a little bit
more about in another video.

00:01:08.557 --> 00:01:10.041
Now the Articles of Confederation

00:01:10.041 --> 00:01:13.874
had a very strong sense
of limited government.

00:01:16.657 --> 00:01:19.307
In fact, you could think of
the Articles of Confederation

00:01:19.307 --> 00:01:20.953
as being a little bit more

00:01:20.953 --> 00:01:24.036
like a loose confederation of states,

00:01:27.042 --> 00:01:31.209
where each state had one vote
in the legislative branch,

00:01:32.666 --> 00:01:34.417
the branch that makes laws,

00:01:34.417 --> 00:01:38.084
and they had to really
agree on most things,

00:01:39.174 --> 00:01:42.565
nine out of 13 for most legislation

00:01:42.565 --> 00:01:44.613
and unanimous agreement

00:01:44.613 --> 00:01:47.083
for any kind of amendments to this system.

00:01:47.083 --> 00:01:49.616
And I think it's clear why the founders

00:01:49.616 --> 00:01:52.592
first went with this system
of limited government

00:01:52.592 --> 00:01:56.908
because they had just
revolted against a monarchy.

00:01:56.908 --> 00:01:59.017
They thought of the states as being

00:01:59.017 --> 00:02:02.508
in what they call just kind
of a league of friendship,

00:02:02.508 --> 00:02:04.025
can almost see it as being similar

00:02:04.025 --> 00:02:06.359
to the European Union today,

00:02:06.359 --> 00:02:10.276
independent nations who
do some things together

00:02:11.149 --> 00:02:14.517
for foreign policy reasons
and economic reasons.

00:02:14.517 --> 00:02:17.691
Now by the late 1780s,
it was becoming clear

00:02:17.691 --> 00:02:21.200
that the Articles of
Confederation were not working.

00:02:21.200 --> 00:02:23.323
With such a weak central government,

00:02:23.323 --> 00:02:25.113
it was really hard to get things done.

00:02:25.113 --> 00:02:26.492
They couldn't raise taxes.

00:02:26.492 --> 00:02:28.975
They couldn't raise a military.

00:02:28.975 --> 00:02:30.966
Some states were even putting taxes

00:02:30.966 --> 00:02:33.137
on the goods of other states.

00:02:33.137 --> 00:02:37.054
So in 1787, delegates
from 12 of the 13 states,

00:02:38.158 --> 00:02:40.831
Rhode Island did not
participate because Rhode Island

00:02:40.831 --> 00:02:43.051
was not a big fan of central government,

00:02:43.051 --> 00:02:44.799
came together in Philadelphia,

00:02:44.799 --> 00:02:46.516
in the same place where they had signed

00:02:46.516 --> 00:02:48.425
the Declaration of Independence,

00:02:48.425 --> 00:02:52.383
to think about how to revise
the Articles of Confederation.

00:02:52.383 --> 00:02:54.724
And some very notable figures were there.

00:02:54.724 --> 00:02:57.141
George Washington is one of them.

00:02:57.141 --> 00:02:59.733
See Ben Franklin over here,

00:02:59.733 --> 00:03:01.233
and James Madison.

00:03:02.748 --> 00:03:05.384
Although some people that
you might have expected

00:03:05.384 --> 00:03:08.982
to be at the Constitutional
Convention were not,

00:03:08.982 --> 00:03:10.982
namely, Thomas Jefferson

00:03:12.807 --> 00:03:13.974
and John Adams

00:03:15.417 --> 00:03:19.584
who were out of the country
being diplomats at the time.

00:03:20.420 --> 00:03:22.775
So the delegates at the
Constitutional Convention

00:03:22.775 --> 00:03:26.041
have a pretty difficult problem to solve.

00:03:26.041 --> 00:03:30.433
They want to have a
stronger central government,

00:03:30.433 --> 00:03:32.316
one that can get things done,

00:03:32.316 --> 00:03:35.177
make sure the states play well together,

00:03:35.177 --> 00:03:37.600
raise armies, raise taxes,

00:03:37.600 --> 00:03:41.249
but they don't want a central
government that is too strong

00:03:41.249 --> 00:03:44.350
because they just escaped from monarchy.

00:03:44.350 --> 00:03:48.392
They don't wanna recreate
monarchy in the United States.

00:03:48.392 --> 00:03:51.878
So they're looking for
a very delicate balance

00:03:51.878 --> 00:03:54.932
of a government strong
enough to get things done

00:03:54.932 --> 00:03:58.265
but not so strong as to promote tyranny.

00:03:59.755 --> 00:04:03.075
Now even though the delegates
were supposed to be revising

00:04:03.075 --> 00:04:04.932
the Articles of Confederation,

00:04:04.932 --> 00:04:06.825
some people had, in secret,

00:04:06.825 --> 00:04:10.774
been considering completely throwing out

00:04:10.774 --> 00:04:13.492
the Articles of Confederation
and starting anew.

00:04:13.492 --> 00:04:15.817
But one of the biggest
hurdles they had to solve

00:04:15.817 --> 00:04:19.926
was what would a new sort
of legislature look like.

00:04:19.926 --> 00:04:22.002
So the Virginia delegates

00:04:22.002 --> 00:04:25.675
suggested a plan for the legislature,

00:04:25.675 --> 00:04:27.758
it's the law-making body,

00:04:29.816 --> 00:04:31.816
that would be bicameral,

00:04:33.192 --> 00:04:35.442
means two room or two house

00:04:36.425 --> 00:04:38.792
from bi, meaning two,

00:04:38.792 --> 00:04:41.042
and camera, Latin for room.

00:04:43.224 --> 00:04:47.064
And their idea was that
there would be a lower house,

00:04:47.064 --> 00:04:51.231
similar to the House of
Commons in English Parliament,

00:04:53.485 --> 00:04:56.567
that would be directly elected

00:04:56.567 --> 00:05:00.300
as individuals would vote
for the representatives

00:05:00.300 --> 00:05:04.396
but the number of representatives
that each state would get

00:05:04.396 --> 00:05:07.479
would be decided by their population.

00:05:08.766 --> 00:05:12.933
Now Virginia was the largest
state by population by far

00:05:14.325 --> 00:05:15.924
and so, this plan would have worked out

00:05:15.924 --> 00:05:18.115
pretty well for them because
they would have gotten

00:05:18.115 --> 00:05:21.269
the largest proportion of representatives.

00:05:21.269 --> 00:05:25.542
Small states like Delaware,
and Georgia, Rhode Island,

00:05:25.542 --> 00:05:29.658
would have very few representatives
indeed, comparatively.

00:05:29.658 --> 00:05:32.991
They also wanted to have an upper house,

00:05:33.850 --> 00:05:37.866
similar to the House of Lords
in the British Parliament,

00:05:37.866 --> 00:05:39.866
which would be appointed

00:05:42.275 --> 00:05:44.599
by state legislatures.

00:05:44.599 --> 00:05:46.342
But just like the lower house,

00:05:46.342 --> 00:05:48.192
the number representatives

00:05:48.192 --> 00:05:51.777
would also be determined by population.

00:05:51.777 --> 00:05:53.625
Now as you can imagine, the small sates

00:05:53.625 --> 00:05:57.758
were not big fans of having representation

00:05:57.758 --> 00:06:00.082
based just on population,

00:06:00.082 --> 00:06:02.616
so they came back with a different plan.

00:06:02.616 --> 00:06:04.850
This was called the New Jersey Plan.

00:06:04.850 --> 00:06:08.624
So the little states said,
"All right, the Virginia Plan

00:06:08.624 --> 00:06:11.777
"gives way too much
power to the big states.

00:06:11.777 --> 00:06:15.583
"We want an equal voice in legislation."

00:06:15.583 --> 00:06:16.850
So the New Jersey Plan,

00:06:16.850 --> 00:06:19.000
much like the Articles of Confederation,

00:06:19.000 --> 00:06:20.083
gave one vote

00:06:22.830 --> 00:06:24.580
to each of the states

00:06:27.043 --> 00:06:29.043
so that the small states

00:06:30.574 --> 00:06:34.925
would have the same
representation in Congress

00:06:34.925 --> 00:06:36.490
as the large states

00:06:36.490 --> 00:06:39.090
and their plan was for a single chamber

00:06:39.090 --> 00:06:41.257
or unicameral legislature.

00:06:43.892 --> 00:06:45.400
So this really wasn't much different

00:06:45.400 --> 00:06:47.208
from the Articles of Confederation at all.

00:06:47.208 --> 00:06:50.132
So how did the delegates
resolve this issue

00:06:50.132 --> 00:06:54.082
of how to balance the
voices of large states

00:06:54.082 --> 00:06:56.091
with large populations

00:06:56.091 --> 00:06:59.363
with small states that
had small populations?

00:06:59.363 --> 00:07:03.407
Because in a situation where all states

00:07:03.407 --> 00:07:05.432
have an equal number of votes,

00:07:05.432 --> 00:07:07.300
like in the New Jersey Plan,

00:07:07.300 --> 00:07:10.349
the 60,000 residents of Delaware

00:07:10.349 --> 00:07:11.924
could have as much say

00:07:11.924 --> 00:07:16.245
as the almost 700,000
residents of Virginia.

00:07:16.245 --> 00:07:18.288
Meaning that the people
who live in Delaware

00:07:18.288 --> 00:07:20.649
were in fact more powerful.

00:07:20.649 --> 00:07:23.682
But at the same time, you wouldn't want it

00:07:23.682 --> 00:07:26.336
so that people living in the larger states

00:07:26.336 --> 00:07:28.832
could get their way all the time.

00:07:28.832 --> 00:07:30.408
What if the people in Delaware

00:07:30.408 --> 00:07:32.666
had a very legitimate concern

00:07:32.666 --> 00:07:34.657
that those in Virginia didn't share?

00:07:34.657 --> 00:07:36.100
It would be impossible

00:07:36.100 --> 00:07:39.400
to get all of these
states to agree, to amend,

00:07:39.400 --> 00:07:42.100
or replace the Articles of Confederation

00:07:42.100 --> 00:07:44.897
if some of them felt like their interests

00:07:44.897 --> 00:07:47.075
aren't being taken into account at all.

00:07:47.075 --> 00:07:50.325
So to solve this issue of how to weight

00:07:51.558 --> 00:07:53.683
the representation of the states,

00:07:53.683 --> 00:07:54.715
the delegates came up

00:07:54.715 --> 00:07:56.833
with what's called the Great Compromise

00:07:56.833 --> 00:08:00.166
or sometimes the Connecticut Compromise.

00:08:01.042 --> 00:08:04.933
And in a way, what they did
was combine these two plans.

00:08:04.933 --> 00:08:08.021
They made a legislative branch

00:08:08.021 --> 00:08:10.521
that was bicameral, two house,

00:08:12.892 --> 00:08:15.475
with a lower and an upper house

00:08:17.599 --> 00:08:21.766
and this lower house become
the House of Representatives,

00:08:24.792 --> 00:08:28.375
where each state would
have representatives

00:08:29.725 --> 00:08:32.558
in proportion to their population.

00:08:33.616 --> 00:08:36.916
So states that have large populations

00:08:36.916 --> 00:08:38.582
have more representatives.

00:08:38.582 --> 00:08:42.749
States with small populations
have fewer representatives.

00:08:44.549 --> 00:08:46.184
And those representatives

00:08:46.184 --> 00:08:49.908
would be directly elected by the people.

00:08:49.908 --> 00:08:51.233
Now, and this time,

00:08:51.233 --> 00:08:54.566
the people was a fairly small proportion

00:08:55.724 --> 00:08:57.141
to vote in 1790s.

00:08:58.816 --> 00:09:00.899
You had to be a white man

00:09:02.007 --> 00:09:04.491
with fairly significant property,

00:09:04.491 --> 00:09:07.183
so it wasn't full suffrage.

00:09:07.183 --> 00:09:10.408
It wasn't event full
suffrage for white men,

00:09:10.408 --> 00:09:13.491
but these folks were elected by vote.

00:09:15.350 --> 00:09:19.183
And then this upper house
would be the Senate.

00:09:20.084 --> 00:09:24.251
And in the Senate, each state
would have two senators,

00:09:26.524 --> 00:09:28.607
regardless of their size,

00:09:30.007 --> 00:09:33.840
so that as legislation
moved through Congress,

00:09:35.080 --> 00:09:39.124
first from the lower house,
where it would be approved

00:09:39.124 --> 00:09:42.599
and if approved, sent to the upper house,

00:09:42.599 --> 00:09:45.700
there, all states would
have an equal voice

00:09:45.700 --> 00:09:48.232
in whether legislation was passed.

00:09:48.232 --> 00:09:49.682
And in this upper house,

00:09:49.682 --> 00:09:52.467
the senators would not be directly elected

00:09:52.467 --> 00:09:54.134
but rather appointed

00:09:56.708 --> 00:09:58.450
by state legislatures.

00:09:58.450 --> 00:10:02.425
And in fact, senators were
appointed into the 20th century.

00:10:02.425 --> 00:10:05.507
Now the Great Compromise
wasn't the only compromise made

00:10:05.507 --> 00:10:07.532
at the Constitutional Convention.

00:10:07.532 --> 00:10:09.092
They made a number more

00:10:09.092 --> 00:10:10.915
and we'll talk more about those

00:10:10.915 --> 00:10:14.249
and about the other two
branches of the government

00:10:14.249 --> 00:10:15.324
in the next video.


The Articles of Confederation

WEBVTT
Kind: captions
Language: en

00:00:00.000 --> 00:00:02.954
- [Instructor] Hey, this is
Kim and I'm here with Leah,

00:00:02.954 --> 00:00:06.015
Khan Academy's US Government
and Politics Fellow.

00:00:06.015 --> 00:00:06.848
Welcome Leah.

00:00:06.848 --> 00:00:07.681
- [Leah] How's it going?

00:00:07.681 --> 00:00:08.986
- [Kim] Alright, so we're talking about

00:00:08.986 --> 00:00:10.816
the Articles of Confederation,

00:00:10.816 --> 00:00:12.647
which I think many people don't realize

00:00:12.647 --> 00:00:15.218
was the first constitution
of the United States

00:00:15.218 --> 00:00:19.192
before the one that we
have now since 1789.

00:00:19.192 --> 00:00:21.076
So could you take us through a little bit

00:00:21.076 --> 00:00:23.539
what the Articles of Confederation were

00:00:23.539 --> 00:00:26.818
and the context in which
we first brought them on

00:00:26.818 --> 00:00:28.142
as a governmental system?

00:00:28.142 --> 00:00:30.167
- [Leah] Sure, so I think
the most important thing

00:00:30.167 --> 00:00:31.818
to understand about the
Articles of Confederation

00:00:31.818 --> 00:00:33.821
and why we would talk about this

00:00:33.821 --> 00:00:36.883
is because one of the
biggest debates that we have

00:00:36.883 --> 00:00:39.098
in our history is about
the balance of power

00:00:39.098 --> 00:00:42.484
between the federal government
and state governments.

00:00:42.484 --> 00:00:45.274
When the Articles of
Confederation were first created,

00:00:45.274 --> 00:00:47.290
it was in the middle of
the American Revolution.

00:00:47.290 --> 00:00:52.107
They were created in 1777,
and so the question becomes,

00:00:52.107 --> 00:00:53.896
well, how can we run a government

00:00:53.896 --> 00:00:56.703
that looks as different
from monarchy as possible?

00:00:56.703 --> 00:00:59.781
- [Kim] Right, so they're
trying to run away from the past

00:00:59.781 --> 00:01:02.325
that they're getting away
from in the Revolutionary War

00:01:02.325 --> 00:01:05.265
and trying to create a separate government

00:01:05.265 --> 00:01:07.769
that doesn't have any of those abuses

00:01:07.769 --> 00:01:10.124
that they are rebelling against.

00:01:10.124 --> 00:01:11.500
- [Leah] If they're running
away from a monarchy,

00:01:11.500 --> 00:01:12.486
what they're running towards

00:01:12.486 --> 00:01:14.466
is what we would call limited government.

00:01:14.466 --> 00:01:15.299
- [Kim] OK.

00:01:15.299 --> 00:01:16.291
- [Leah] So their central government,

00:01:16.291 --> 00:01:18.878
which is synonymous with
a federal government,

00:01:18.878 --> 00:01:20.650
the central government is actually

00:01:20.650 --> 00:01:22.400
really, really, really small.

00:01:22.400 --> 00:01:23.233
- [Kim] OK.

00:01:23.233 --> 00:01:24.396
- [Leah] They don't have
an executive branch.

00:01:24.396 --> 00:01:25.731
They only have Congress.

00:01:25.731 --> 00:01:27.670
They don't even have a judicial branch.

00:01:27.670 --> 00:01:31.295
So Congress is made up of all 13 states.

00:01:31.295 --> 00:01:33.327
Every state had one representative.

00:01:33.327 --> 00:01:34.160
- [Kim] OK.

00:01:34.160 --> 00:01:36.959
- [Leah] In order to change
the Articles of Confederation,

00:01:36.959 --> 00:01:38.759
if they wanted to pass an amendment,

00:01:38.759 --> 00:01:42.598
they had to get unanimous
consent from all 13 states.

00:01:42.598 --> 00:01:44.319
- [Kim] OK, so they're trying to make sure

00:01:44.319 --> 00:01:48.642
that all of the states
are represented equally,

00:01:48.642 --> 00:01:52.293
but that also sounds like it
would have a lot of hurdles

00:01:52.293 --> 00:01:56.408
to overcome when it comes
to getting consensus.

00:01:56.408 --> 00:01:58.638
- [Leah] Yeah, for laws,
you had to get nine

00:01:58.638 --> 00:02:01.737
out of 13 states to actually pass a law.

00:02:01.737 --> 00:02:04.893
So if you can imagine, if
you're in a room of 13 people

00:02:04.893 --> 00:02:07.997
and you all have to agree
on one pizza topping

00:02:07.997 --> 00:02:10.170
for the rest of your lives.
(Kim laughs)

00:02:10.170 --> 00:02:11.928
It would be almost impossible, right?

00:02:11.928 --> 00:02:14.819
- [Kim] Wow, OK, alright,
so it sounds like

00:02:14.819 --> 00:02:18.198
there are some problems with
the Articles of Confederation,

00:02:18.198 --> 00:02:21.501
but did they do anything good
for us in this early period?

00:02:21.501 --> 00:02:22.870
- [Leah] Yeah, so the biggest thing

00:02:22.870 --> 00:02:26.652
is that it unites all 13
colonies who are now states

00:02:26.652 --> 00:02:28.934
under one government.

00:02:28.934 --> 00:02:29.767
- [Kim] OK.

00:02:29.767 --> 00:02:31.158
- [Leah] This government is able to pass

00:02:31.158 --> 00:02:33.745
a really favorable treaty with Britain

00:02:33.745 --> 00:02:37.968
and end the Revolutionary War
in 1783, the Treaty of Paris.

00:02:37.968 --> 00:02:39.804
- [Kim] OK, so this is
kind of the government

00:02:39.804 --> 00:02:42.110
that gets us through the revolutionary war

00:02:42.110 --> 00:02:46.856
and is with us when we
first start in the 1780s.

00:02:46.856 --> 00:02:50.011
- [Leah] Yeah, and one another
specific law that they pass

00:02:50.011 --> 00:02:53.224
is the Northwest Ordinance of 1787

00:02:53.224 --> 00:02:56.226
and this Northwest
Ordinance kinda tells us

00:02:56.226 --> 00:02:59.413
how we are going to expand as we move west

00:02:59.413 --> 00:03:01.047
and what are we going
to do with that land,

00:03:01.047 --> 00:03:03.119
and that's a really important idea

00:03:03.119 --> 00:03:05.978
when we're moving
forward with our country.

00:03:05.978 --> 00:03:08.571
The only problem is with the
Articles of Confederation

00:03:08.571 --> 00:03:11.347
is there is a lot of things
that we still have to figure out

00:03:11.347 --> 00:03:13.535
as we're growing, there's
a lot of growing pains.

00:03:13.535 --> 00:03:16.126
- [Kim] So what led the early government

00:03:16.126 --> 00:03:18.085
of the United States to realize

00:03:18.085 --> 00:03:20.873
that they wanted to abandon
these Articles of Confederation

00:03:20.873 --> 00:03:23.569
in favor of a different constitution?

00:03:23.569 --> 00:03:26.214
- [Leah] So the inciting
incident is Shays' Rebellion.

00:03:26.214 --> 00:03:29.006
It happens in Massachusets,
and it's a group of farmers

00:03:29.006 --> 00:03:31.492
led by this guy named Daniel Shays.

00:03:31.492 --> 00:03:33.879
What's happening is that
we had just gotten out of

00:03:33.879 --> 00:03:35.861
the Revolutionary War,
and a lot of of the people

00:03:35.861 --> 00:03:37.410
who had fought in the Revolutionary War

00:03:37.410 --> 00:03:40.050
still hadn't gotten
payment for their duty.

00:03:40.050 --> 00:03:43.376
They also were experiencing
really high state taxes,

00:03:43.376 --> 00:03:47.638
so Daniel Shays and these farmers
are very upset, obviously,

00:03:47.638 --> 00:03:50.874
in their wanting their money
and so they started rebelling,

00:03:50.874 --> 00:03:53.447
but the problem is, with the
way that the central government

00:03:53.447 --> 00:03:57.147
that was built, first,
Congress had no ability

00:03:57.147 --> 00:03:59.389
to levy or collect taxes.

00:03:59.389 --> 00:04:00.607
If they couldn't collect taxes,

00:04:00.607 --> 00:04:04.547
they had no ability to actually
pay back their farmers.

00:04:04.547 --> 00:04:07.413
Along with that, they
didn't have any money

00:04:07.413 --> 00:04:10.648
to create a military, so each
state had their own militia,

00:04:10.648 --> 00:04:13.478
but the United States
as a whole as a country

00:04:13.478 --> 00:04:16.123
did not have a military to
suppress this rebellion.

00:04:16.123 --> 00:04:18.457
So on both ends, we are in

00:04:18.457 --> 00:04:21.690
a really bad situation politically.

00:04:21.690 --> 00:04:23.843
- [Kim] Wow, so, there's this moment

00:04:23.843 --> 00:04:27.758
where you find armed rebellion
against the United States

00:04:27.758 --> 00:04:31.195
for a lack of money and the US government

00:04:31.195 --> 00:04:32.957
finds that it can't raise money

00:04:32.957 --> 00:04:35.907
and it can't raise an army
to put down this rebellion.

00:04:35.907 --> 00:04:39.393
- [Leah] Exactly, and so there
is this fear immediately.

00:04:39.393 --> 00:04:41.305
And what we see is a lot
of the founding fathers

00:04:41.305 --> 00:04:44.044
that we know and really respect
today like George Washington

00:04:44.044 --> 00:04:47.177
and Ben Franklin and Alexander
Hamilton and James Madison,

00:04:47.177 --> 00:04:49.678
they get together and they
say this is a problem,

00:04:49.678 --> 00:04:51.685
we need to change what we have,

00:04:51.685 --> 00:04:53.728
and this leads to the
Constitutional Convention

00:04:53.728 --> 00:04:56.239
in which we draft our second constitution.

00:04:56.239 --> 00:04:58.964
- [Kim] Right, yeah, so in 1787,

00:04:58.964 --> 00:05:02.301
the leaders of the United
States get together and say,

00:05:02.301 --> 00:05:04.676
alright, the Articles of
Confederation aren't working.

00:05:04.676 --> 00:05:06.884
We're gonna need a stronger
central government,

00:05:06.884 --> 00:05:09.500
even though we were trying
to get away from the monarchy

00:05:09.500 --> 00:05:11.447
and now let's think of something

00:05:11.447 --> 00:05:13.628
that's going to work a
little bit better for us.


The Declaration of Independence

WEBVTT
Kind: captions
Language: en

00:00:00.977 --> 00:00:03.240
- [Kim] On July fourth, 1776,

00:00:03.240 --> 00:00:05.963
the delegates to the
Second Continental Congress

00:00:05.963 --> 00:00:09.297
approved the Declaration of Independence.

00:00:09.297 --> 00:00:11.573
We know parts of it very well.

00:00:11.573 --> 00:00:12.479
For example,

00:00:12.479 --> 00:00:15.861
"We hold these truths to be self-evident,"

00:00:15.861 --> 00:00:17.763
"that all men are created equal."

00:00:17.763 --> 00:00:20.280
The Declaration of
Independence was really the

00:00:20.280 --> 00:00:22.237
point of no return

00:00:22.237 --> 00:00:25.090
for the young United States of America,

00:00:25.090 --> 00:00:27.866
making an appeal to the rest of the world,

00:00:27.866 --> 00:00:31.485
to say that their time as a
colony of the United Kingdom

00:00:31.485 --> 00:00:32.318
had ended.

00:00:33.191 --> 00:00:36.853
The principle author of the
Declaration of Independence

00:00:36.853 --> 00:00:39.853
was this man here, Thomas Jefferson.

00:00:40.788 --> 00:00:43.586
I tried to find a picture
of him as a young man.

00:00:43.586 --> 00:00:45.422
In fact, at the time,

00:00:45.422 --> 00:00:48.991
he was about 10 years younger
than you even see him here.

00:00:48.991 --> 00:00:49.824
He was 33.

00:00:52.421 --> 00:00:57.026
Where did young Thomas
Jefferson get all of the ideas

00:00:57.026 --> 00:01:00.693
that he expressed in the
Declaration of Independence,

00:01:00.693 --> 00:01:02.729
and what happened to those ideas,

00:01:02.729 --> 00:01:04.584
once he put them down on paper?

00:01:04.584 --> 00:01:05.567
In this video,

00:01:05.567 --> 00:01:07.705
I'd like to explore some of the origins

00:01:07.705 --> 00:01:11.863
and effects of the
Declaration of Independence.

00:01:11.863 --> 00:01:15.670
We often think that the
Revolutionary War started

00:01:15.670 --> 00:01:18.528
with the Declaration of Independence.

00:01:18.528 --> 00:01:20.965
We think of 1776 as being this

00:01:20.965 --> 00:01:24.369
opening moment of the Revolution.

00:01:24.369 --> 00:01:28.762
In fact, parts of the
Revolutionary War had been going on

00:01:28.762 --> 00:01:30.225
for some time.

00:01:30.225 --> 00:01:33.876
It was in 1765, more
than a decade earlier,

00:01:33.876 --> 00:01:37.543
that some of the first
unrest over taxation,

00:01:39.599 --> 00:01:42.766
specifically the Stamp Act, had begun.

00:01:43.935 --> 00:01:47.132
In 1773, the famous Boston Tea Party,

00:01:47.132 --> 00:01:51.299
when a group of colonists
dumped over 300 crates of tea

00:01:52.329 --> 00:01:55.313
into Boston Harbor had happened.

00:01:55.313 --> 00:01:59.480
In 1775, over a year before the
Declaration of Independence,

00:02:01.644 --> 00:02:03.365
the first shots

00:02:03.365 --> 00:02:07.532
of the Revolutionary War had
taken place outside Boston,

00:02:08.924 --> 00:02:11.749
at the towns of Lexington and Concord.

00:02:11.749 --> 00:02:15.445
By the time the delegates
had met in Philadelphia,

00:02:15.445 --> 00:02:18.254
the Revolutionary War
had been a shooting war

00:02:18.254 --> 00:02:20.212
for more than a year.

00:02:20.212 --> 00:02:22.879
Why was it that in July of 1776,

00:02:24.704 --> 00:02:28.762
the delegates finally made the
Declaration of Independence?

00:02:28.762 --> 00:02:30.640
The primary reason that
they did it at this time,

00:02:30.640 --> 00:02:33.057
was because they wanted help,

00:02:35.998 --> 00:02:40.165
and they were particularly
eager to get the assistance

00:02:41.534 --> 00:02:43.531
of the nation of France,

00:02:43.531 --> 00:02:48.322
which had been a long time
enemy of the United Kingdom,

00:02:48.322 --> 00:02:51.899
and the delegates really knew that the new

00:02:51.899 --> 00:02:55.165
United States of America
would have no hope of winning

00:02:55.165 --> 00:02:59.332
a war against a massive imperial
power like Great Britain,

00:03:00.168 --> 00:03:04.335
without the help of another
world power, such as France.

00:03:05.234 --> 00:03:07.498
In a way, what Jefferson was doing in the

00:03:07.498 --> 00:03:09.066
Declaration of Independence,

00:03:09.066 --> 00:03:12.983
wasn't so much declaring,
but rather explaining

00:03:15.361 --> 00:03:19.121
why the states were declaring
themselves independent,

00:03:19.121 --> 00:03:21.767
with the hope that they
could get the sympathy

00:03:21.767 --> 00:03:25.364
and the help of the
international community.

00:03:25.364 --> 00:03:28.425
Let's read some of the
Declaration of Independence.

00:03:28.425 --> 00:03:31.760
I know that this is a
gigantic block of text here,

00:03:31.760 --> 00:03:33.126
but bear with me.

00:03:33.126 --> 00:03:34.664
We'll grow through it fairly quickly.

00:03:34.664 --> 00:03:36.993
"The unanimous Declaration
of the thirteen"

00:03:36.993 --> 00:03:39.536
"United States of America."

00:03:39.536 --> 00:03:42.267
You'll notice here that
he specifically points out

00:03:42.267 --> 00:03:45.130
that there are 13 United States.

00:03:45.130 --> 00:03:49.126
This is important, because
it gives you a sense that

00:03:49.126 --> 00:03:52.802
they aren't really thinking
of the individual former

00:03:52.802 --> 00:03:56.627
colonies, now states,
as one larger country,

00:03:56.627 --> 00:04:00.026
but rather as a collection of states,

00:04:00.026 --> 00:04:02.883
a confederation of allied states,

00:04:02.883 --> 00:04:05.732
instead of a single nation.

00:04:05.732 --> 00:04:08.932
"When in the course of human
events it becomes necessary"

00:04:08.932 --> 00:04:12.353
"for one people to dissolve
the political bands"

00:04:12.353 --> 00:04:14.534
"which have connected them with another"

00:04:14.534 --> 00:04:17.382
"and to assume among the
powers of the Earth,"

00:04:17.382 --> 00:04:18.667
"the separate and equal"

00:04:18.667 --> 00:04:20.688
"station to which the laws of nature,"

00:04:20.688 --> 00:04:22.689
"and of nature's God entitle them."

00:04:22.689 --> 00:04:25.268
"A decent respect to the
opinions of mankind requires"

00:04:25.268 --> 00:04:28.321
"that they should declare
the causes which impel them"

00:04:28.321 --> 00:04:30.199
"to the separation."

00:04:30.199 --> 00:04:33.187
Here's this explanation part,

00:04:33.187 --> 00:04:37.703
saying, we feel it necessary
to explain why we want to

00:04:37.703 --> 00:04:39.631
separate from Great Britain.

00:04:39.631 --> 00:04:43.121
"We hold these truths to be self-evident."

00:04:43.121 --> 00:04:45.205
"That all men are created equal,"

00:04:45.205 --> 00:04:47.153
"that they are endowed by their Creator"

00:04:47.153 --> 00:04:49.216
"with certain unalienable rights,"

00:04:49.216 --> 00:04:51.335
"that among these are life, liberty,"

00:04:51.335 --> 00:04:53.829
"and the pursuit of happiness."

00:04:53.829 --> 00:04:55.966
"That to secure these rights,"

00:04:55.966 --> 00:04:58.067
"governments are instituted among men,"

00:04:58.067 --> 00:04:59.895
"deriving their just powers from the"

00:04:59.895 --> 00:05:02.394
"consent of the governed."

00:05:02.394 --> 00:05:05.357
"That whenever any form of
government becomes destructive"

00:05:05.357 --> 00:05:06.584
"of these ends,"

00:05:06.584 --> 00:05:09.162
"it is the right of the people to alter,"

00:05:09.162 --> 00:05:10.723
"or to abolish it,"

00:05:10.723 --> 00:05:12.363
"and to institute new government,"

00:05:12.363 --> 00:05:15.892
"laying its foundation on such
principles and organizing"

00:05:15.892 --> 00:05:17.773
"its powers in such form,"

00:05:17.773 --> 00:05:20.262
"as to them shall seem
most likely to effect"

00:05:20.262 --> 00:05:22.605
"their safety and happiness."

00:05:22.605 --> 00:05:26.350
I think this might be the
most important passage

00:05:26.350 --> 00:05:28.298
of the Declaration of Independence,

00:05:28.298 --> 00:05:29.891
and let me tell you why.

00:05:29.891 --> 00:05:31.806
In this paragraph, you can really see the

00:05:31.806 --> 00:05:35.973
influence of the Enlightenment
on Jefferson's thought.

00:05:37.321 --> 00:05:41.488
The Enlightenment was a
period in the 1600s and 1700s,

00:05:42.981 --> 00:05:45.314
when people began to explore

00:05:48.677 --> 00:05:50.510
scientific observation

00:05:52.187 --> 00:05:53.104
and reason.

00:05:54.845 --> 00:05:57.995
They became more interested
in observing the world

00:05:57.995 --> 00:06:01.376
around them, and trying to make

00:06:01.376 --> 00:06:04.543
reasoned arguments from what they saw,

00:06:05.465 --> 00:06:10.277
as compared to accepting
the religious explanations

00:06:10.277 --> 00:06:12.291
for how the world worked.

00:06:12.291 --> 00:06:14.547
During the Enlightenment,
many philosophers began to

00:06:14.547 --> 00:06:16.797
rethink government as well,

00:06:18.998 --> 00:06:22.482
and of questioning whether
the governmental system

00:06:22.482 --> 00:06:27.085
in Europe and other places
was the right system.

00:06:27.085 --> 00:06:28.829
There was one philosopher, in particular,

00:06:28.829 --> 00:06:32.643
who really captured
Jefferson's imagination,

00:06:32.643 --> 00:06:35.116
and his name was John Locke.

00:06:35.116 --> 00:06:38.283
John Lcoke was an English philosopher,

00:06:40.048 --> 00:06:42.139
who had lived in the 1600s,

00:06:42.139 --> 00:06:45.162
and he wrote a book that
had really influenced

00:06:45.162 --> 00:06:48.581
Jefferson and many thinkers
in this time period,

00:06:48.581 --> 00:06:51.498
called Two Treatises on Government.

00:06:54.850 --> 00:06:59.017
There are two really important
points in Locke's work.

00:07:00.035 --> 00:07:02.952
One was the idea of natural rights.

00:07:05.859 --> 00:07:09.095
What Locke meant by natural rights,

00:07:09.095 --> 00:07:12.771
are rights that were endowed by nature,

00:07:12.771 --> 00:07:16.169
that all people were born with.

00:07:16.169 --> 00:07:19.199
If you think about Europe
in this time period,

00:07:19.199 --> 00:07:22.699
there was a sense that
some people were born with

00:07:22.699 --> 00:07:24.816
more rights than others.

00:07:24.816 --> 00:07:28.983
In fact, there was the idea
of the Divine Right of Kings,

00:07:29.859 --> 00:07:31.026
that the king,

00:07:33.517 --> 00:07:35.434
or monarch of any sort,

00:07:36.319 --> 00:07:40.152
had been born the king
because God wanted that

00:07:42.354 --> 00:07:43.604
person to rule.

00:07:44.587 --> 00:07:46.892
Locke rejects that.

00:07:46.892 --> 00:07:49.483
He says when people are born,

00:07:49.483 --> 00:07:51.478
they're all born the same,

00:07:51.478 --> 00:07:56.460
and they all have rights
that can't be given away,

00:07:56.460 --> 00:07:58.593
that are unalienable,

00:07:58.593 --> 00:08:00.722
and those are life,

00:08:00.722 --> 00:08:01.555
liberty,

00:08:03.279 --> 00:08:04.362
and property.

00:08:05.735 --> 00:08:08.807
Sounds pretty familiar, doesn't it?

00:08:08.807 --> 00:08:11.688
The other important idea that Locke had,

00:08:11.688 --> 00:08:14.688
was the idea of the Social Contract.

00:08:17.263 --> 00:08:20.304
The idea of the Social Contract was that,

00:08:20.304 --> 00:08:23.162
government came from the people,

00:08:23.162 --> 00:08:26.382
that society members got together,

00:08:26.382 --> 00:08:30.689
and agreed on what the forms
of government should be,

00:08:30.689 --> 00:08:33.852
so that the only just government,

00:08:33.852 --> 00:08:36.739
was one that took into
account the opinions

00:08:36.739 --> 00:08:39.594
of the people who live within it.

00:08:39.594 --> 00:08:41.288
You see that here, too.

00:08:41.288 --> 00:08:43.530
"Governments are instituted among men,"

00:08:43.530 --> 00:08:45.780
"deriving their just powers from"

00:08:45.780 --> 00:08:47.857
"the consent of the governed."

00:08:47.857 --> 00:08:50.434
There's a third thing
that Locke suggests that

00:08:50.434 --> 00:08:53.003
Jefferson also gets at,

00:08:53.003 --> 00:08:56.126
which is that when
governments become tyrannical,

00:08:56.126 --> 00:08:59.683
when they do not abide
by the Social Contract,

00:08:59.683 --> 00:09:02.933
it is the right of the people to rebel.

00:09:05.501 --> 00:09:08.095
All right, back to the Declaration.

00:09:08.095 --> 00:09:11.017
Most of the rest of the
Declaration is just a list

00:09:11.017 --> 00:09:14.454
of grievances of what the King has done

00:09:14.454 --> 00:09:17.703
to the colonies that has
made them very angry.

00:09:17.703 --> 00:09:21.144
This is an extremely
abridged list of them.

00:09:21.144 --> 00:09:24.092
I highly recommend you read
the entire Declaration,

00:09:24.092 --> 00:09:26.440
'cause I think it gives
you a really good sense

00:09:26.440 --> 00:09:31.402
of what the colonists were
thinking at this time period.

00:09:31.402 --> 00:09:33.309
Here are some of the highlights.

00:09:33.309 --> 00:09:35.241
Jefferson says that the

00:09:35.241 --> 00:09:38.001
"King has kept among
us, in times of peace,"

00:09:38.001 --> 00:09:41.675
"standing armies without the
consent of our legislatures."

00:09:41.675 --> 00:09:45.678
Those are the British regulars
who have been stationed in

00:09:45.678 --> 00:09:48.178
North America for a long time.

00:09:50.955 --> 00:09:55.299
"For cutting off our trade
with all parts of the world."

00:09:55.299 --> 00:09:59.586
The Navigation Acts,
that said the colonies

00:09:59.586 --> 00:10:02.586
could only trade with Great Britain.

00:10:05.222 --> 00:10:08.845
"For imposing taxes on
us, without our consent."

00:10:08.845 --> 00:10:11.335
The idea of taxation
without representation,

00:10:11.335 --> 00:10:15.168
which really motivated
the colonists to rebel.

00:10:19.983 --> 00:10:22.251
And so forth and so on.

00:10:22.251 --> 00:10:25.955
Now, it's worth noting that the
Declaration of Independence,

00:10:25.955 --> 00:10:30.103
as an explanation hoping
to get France on the side

00:10:30.103 --> 00:10:32.474
of the new United States,

00:10:32.474 --> 00:10:33.998
worked very well.

00:10:33.998 --> 00:10:37.821
The United States allied with France,

00:10:37.821 --> 00:10:41.988
which led them to win the
Revolutionary War in 1783.

00:10:48.083 --> 00:10:50.377
As we close, it's worth pondering,

00:10:50.377 --> 00:10:52.713
what it was that Thomas Jefferson

00:10:52.713 --> 00:10:55.100
really meant by the phrase,

00:10:55.100 --> 00:10:57.479
"All men are created equal."

00:10:57.479 --> 00:11:00.053
The Revolutionary War
didn't abolish slavery

00:11:00.053 --> 00:11:01.469
in the United States.

00:11:01.469 --> 00:11:06.154
In fact, Jefferson himself
owned over 100 enslaved

00:11:06.154 --> 00:11:08.366
people of African descent.

00:11:08.366 --> 00:11:12.092
When Jefferson said, "All
men are created equal,"

00:11:12.092 --> 00:11:15.259
was he thinking only of all white men?

00:11:16.878 --> 00:11:19.834
Was he thinking only of elite white men?

00:11:19.834 --> 00:11:23.125
After all, after the Revolution,

00:11:23.125 --> 00:11:27.292
only a handful of propertied
elite men could vote.

00:11:29.689 --> 00:11:33.022
But then, there's this larger idea here.

00:11:33.940 --> 00:11:38.213
He's saying that your
ordinary man wasn't born

00:11:38.213 --> 00:11:41.489
any different than someone
who was born a king,

00:11:41.489 --> 00:11:44.303
so why should someone who was born black

00:11:44.303 --> 00:11:47.153
be different than someone
who was born white?

00:11:47.153 --> 00:11:50.116
It's hard to imagine how
Jefferson separated those things

00:11:50.116 --> 00:11:51.557
in his mind.

00:11:51.557 --> 00:11:54.229
Certainly, others at the time period,

00:11:54.229 --> 00:11:57.571
realized that there was
an inherent contradiction

00:11:57.571 --> 00:11:59.477
between slavery,

00:11:59.477 --> 00:12:02.229
and also between the rights of women,

00:12:02.229 --> 00:12:05.176
and the idea that all
men are created equal.

00:12:05.176 --> 00:12:07.748
Over time, the rhetoric of the
Declaration of Independence

00:12:07.748 --> 00:12:10.945
would go on to spur many
different movements for

00:12:10.945 --> 00:12:13.455
independence and civil rights.

00:12:13.455 --> 00:12:16.203
The most notable of which
being the French Revolution,

00:12:16.203 --> 00:12:19.536
which took much of its rhetoric from the

00:12:20.556 --> 00:12:22.553
American Revolution.

00:12:22.553 --> 00:12:26.748
Later, in 1848, the first
women's rights movement

00:12:26.748 --> 00:12:29.648
would gather at Seneca Falls, New York,

00:12:29.648 --> 00:12:31.195
and release what they called

00:12:31.195 --> 00:12:34.359
the Declaration of Sentiments.

00:12:34.359 --> 00:12:38.526
Which began, "All men and
women are created equal."

00:12:40.853 --> 00:12:42.902
So Jefferson's ideas here,

00:12:42.902 --> 00:12:45.089
are both deeply radical,

00:12:45.089 --> 00:12:50.053
insisting that ordinary people
are just as good as kings,

00:12:50.053 --> 00:12:51.460
and even more,

00:12:51.460 --> 00:12:55.120
entitled to decide their
own form of government.

00:12:55.120 --> 00:12:58.977
But that Revolution only went so far.

00:12:58.977 --> 00:13:02.394
It didn't change much about the status of

00:13:03.525 --> 00:13:07.217
every day citizens in the United States.

00:13:07.217 --> 00:13:09.782
But it put forward an ideal which

00:13:09.782 --> 00:13:12.191
we've been working toward ever since.


How did Russia begin?

WEBVTT
Kind: captions
Language: en

00:00:00.817 --> 00:00:01.915
- [Instructor] What
we're going to talk about

00:00:01.915 --> 00:00:06.399
in this video are the origins
of the Russian people.

00:00:06.399 --> 00:00:07.794
And in particular, we're
going to talk about

00:00:07.794 --> 00:00:11.711
the eastern Slavs who
not just modern Russians,

00:00:12.603 --> 00:00:16.222
but also Ukrainians and Belorussians

00:00:16.222 --> 00:00:18.222
view as their ancestors.

00:00:19.643 --> 00:00:20.826
So let's think about the world

00:00:20.826 --> 00:00:22.576
in the ninth century.

00:00:23.526 --> 00:00:27.832
The early ninth century, we
see the reign of Charlemagne

00:00:27.832 --> 00:00:30.433
that we talk about in some
depth in other videos.

00:00:30.433 --> 00:00:33.190
As we get into the 10th century,

00:00:33.190 --> 00:00:35.547
you see the reign of Otto the Great,

00:00:35.547 --> 00:00:39.651
Holy Roman Emperor over
the Germanic kingdoms.

00:00:39.651 --> 00:00:44.428
The ninth century is also
the time of Tang China.

00:00:44.428 --> 00:00:47.196
You have the Abbasid Muslim Caliphate

00:00:47.196 --> 00:00:49.570
in control over most of the Middle East

00:00:49.570 --> 00:00:51.381
and North Africa.

00:00:51.381 --> 00:00:53.881
And it is also the Viking age.

00:00:55.579 --> 00:00:58.295
So we have here in this magenta color,

00:00:58.295 --> 00:01:00.590
this would be modern day Sweden,

00:01:00.590 --> 00:01:03.439
but it was also the
home of the Varangians,

00:01:03.439 --> 00:01:07.352
or whom we later would
refer to as the Vikings,

00:01:07.352 --> 00:01:09.947
and we know them to be great seafarers.

00:01:09.947 --> 00:01:11.763
In Western Europe, they're viewed

00:01:11.763 --> 00:01:14.051
as raiders of towns along the coast.

00:01:14.051 --> 00:01:15.442
But you have to remember, these histories

00:01:15.442 --> 00:01:17.844
are often written by
the western Europeans,

00:01:17.844 --> 00:01:19.943
not by the Vikings themselves.

00:01:19.943 --> 00:01:23.096
But they were also known as traders.

00:01:23.096 --> 00:01:25.610
What you see here are
two of the major centers

00:01:25.610 --> 00:01:29.115
of power and trade in the ninth century.

00:01:29.115 --> 00:01:30.791
You have Constantinople,

00:01:30.791 --> 00:01:33.209
the capital of the Byzantine Empire,

00:01:33.209 --> 00:01:35.328
and you have Baghdad, the capital

00:01:35.328 --> 00:01:38.039
of the Abbasid Caliphate.

00:01:38.039 --> 00:01:42.332
You also see these major
waterways in eastern Europe,

00:01:42.332 --> 00:01:45.998
in particular, the Dnieper
and the Volga Rivers.

00:01:45.998 --> 00:01:48.832
And so you have these
significant trade routes

00:01:48.832 --> 00:01:50.458
going from the Baltic Sea

00:01:50.458 --> 00:01:54.006
either via the Dnieper,
crossing the Black Sea

00:01:54.006 --> 00:01:57.356
to Constantinople, or
going from the Baltic

00:01:57.356 --> 00:02:00.512
to the Volga all the
way to the Caspian Sea

00:02:00.512 --> 00:02:03.961
and eventually making
their way to Baghdad.

00:02:03.961 --> 00:02:05.857
And this is well documented.

00:02:05.857 --> 00:02:07.357
There is archeological evidence

00:02:07.357 --> 00:02:10.440
of Viking jewelry along these routes.

00:02:11.381 --> 00:02:15.182
There's evidence of artifacts
from these far off lands

00:02:15.182 --> 00:02:18.818
in Viking territory, and we believe

00:02:18.818 --> 00:02:20.802
what the Varangians traded

00:02:20.802 --> 00:02:25.230
were first they hunted in
this area of northern Europe.

00:02:25.230 --> 00:02:28.863
Now, the people who lived in this area

00:02:28.863 --> 00:02:30.863
were known as the Slavs.

00:02:31.794 --> 00:02:34.487
And there were several
broad groups of Slavs

00:02:34.487 --> 00:02:36.723
that you will hear historians refer to.

00:02:36.723 --> 00:02:38.857
You have the Western Slavs,

00:02:38.857 --> 00:02:41.143
who you could view as
some of the ancestors

00:02:41.143 --> 00:02:44.226
of modern Poles, Czechs, and Slovaks.

00:02:45.774 --> 00:02:46.781
You have the Southern Slavs

00:02:46.781 --> 00:02:48.296
in what would eventually be referred to

00:02:48.296 --> 00:02:50.012
as the Balkans.

00:02:50.012 --> 00:02:51.503
And then you have the Eastern Slavs

00:02:51.503 --> 00:02:55.670
in what will eventually be
Russia, Belarus, and Ukraine.

00:02:57.547 --> 00:03:00.315
Now to be clear, they
weren't unified groups.

00:03:00.315 --> 00:03:02.208
There were many many many, for example,

00:03:02.208 --> 00:03:04.041
Eastern Slavic tribes.

00:03:04.879 --> 00:03:07.377
Our best account of the early history,

00:03:07.377 --> 00:03:11.357
especially in the period as
we get into the ninth century

00:03:11.357 --> 00:03:15.597
comes from what is known as
the Russian Primary Chronicle.

00:03:15.597 --> 00:03:17.705
And keep in mind, this was written

00:03:17.705 --> 00:03:19.599
at a much later period.

00:03:19.599 --> 00:03:23.004
It was written in the early 12th century.

00:03:23.004 --> 00:03:26.968
It is sometimes ascribed
to the Kievan monk Nestor.

00:03:26.968 --> 00:03:29.892
So at previous times, it was
known as Nestor's Chronicles,

00:03:29.892 --> 00:03:32.333
but we don't even have
surviving accounts of this.

00:03:32.333 --> 00:03:34.940
We have surviving accounts
of copies of this,

00:03:34.940 --> 00:03:37.722
or what we believe are copies of this.

00:03:37.722 --> 00:03:39.309
What I'll share is a version

00:03:39.309 --> 00:03:41.249
of the Russian Primary Chronicle

00:03:41.249 --> 00:03:45.144
known as the Laurentian Text from 1377.

00:03:45.144 --> 00:03:47.658
And this is, of course,
an English translation.

00:03:47.658 --> 00:03:49.168
It gives us some of the earliest accounts

00:03:49.168 --> 00:03:52.585
of the relationship between the Varangians

00:03:52.585 --> 00:03:56.561
and the Eastern Slavs
and how what we have come

00:03:56.561 --> 00:03:58.790
to identify as the Russian people

00:03:58.790 --> 00:04:01.594
and the Ukrainian people
and the Belorussian people,

00:04:01.594 --> 00:04:03.677
how they got their start.

00:04:04.834 --> 00:04:06.462
So right before this passage,

00:04:06.462 --> 00:04:10.663
it talks about how the
Varangians tried to get tribute

00:04:10.663 --> 00:04:13.900
from some of the Eastern Slavic tribes.

00:04:13.900 --> 00:04:16.629
And it says the tributaries
of the Varangians

00:04:16.629 --> 00:04:18.940
drove them back beyond the sea

00:04:18.940 --> 00:04:20.851
and, refusing them further tribute,

00:04:20.851 --> 00:04:23.321
set out to govern themselves.

00:04:23.321 --> 00:04:25.664
So they pushed them back
beyond what we now call

00:04:25.664 --> 00:04:29.609
the Baltic Sea, and they
decided to govern themselves.

00:04:29.609 --> 00:04:31.669
There was no law among them.

00:04:31.669 --> 00:04:33.802
Tribe rose against tribe,

00:04:33.802 --> 00:04:36.758
and they began to war one against another.

00:04:36.758 --> 00:04:40.032
They said to themselves,
let us seek a prince

00:04:40.032 --> 00:04:44.099
who may rule over us and
judge us according to the law.

00:04:44.099 --> 00:04:46.093
They accordingly went overseas

00:04:46.093 --> 00:04:48.410
to the Varangian Russes.

00:04:48.410 --> 00:04:50.014
So they went back to the Russes,

00:04:50.014 --> 00:04:53.673
and they said, these warring
Eastern Slavic tribes said,

00:04:53.673 --> 00:04:55.395
our land is great and rich,

00:04:55.395 --> 00:04:57.463
but there is no order in it.

00:04:57.463 --> 00:05:00.794
Come to rule and reign over us.

00:05:00.794 --> 00:05:02.993
They thus selected three brothers.

00:05:02.993 --> 00:05:06.910
The oldest, Rurik, located
himself in Novgorod.

00:05:07.777 --> 00:05:09.136
Right over here.

00:05:09.136 --> 00:05:12.099
Novgorod literally means new town.

00:05:12.099 --> 00:05:13.857
Gorod means town.

00:05:13.857 --> 00:05:15.532
The district of Novgorod

00:05:15.532 --> 00:05:18.282
became known as the land of Rus'.

00:05:19.820 --> 00:05:22.881
So a lot of really
interesting things going on.

00:05:22.881 --> 00:05:25.903
The Varangians, first,
are trying to get tribute

00:05:25.903 --> 00:05:29.815
from these tribes, which is
a way of saying tax them,

00:05:29.815 --> 00:05:33.359
making them subservient to these Vikings.

00:05:33.359 --> 00:05:34.754
And even though these Eastern Slavs

00:05:34.754 --> 00:05:36.373
were able to push them back,

00:05:36.373 --> 00:05:38.291
according to the Primary Chronicle,

00:05:38.291 --> 00:05:40.196
they said hey we need your help.

00:05:40.196 --> 00:05:42.787
We want you to rule over us.

00:05:42.787 --> 00:05:44.839
There's very few times in history

00:05:44.839 --> 00:05:47.554
where people are asking a foreign group

00:05:47.554 --> 00:05:49.672
to rule over them.

00:05:49.672 --> 00:05:51.589
And so this is an interesting question.

00:05:51.589 --> 00:05:55.216
Remember, this history
is written under the rule

00:05:55.216 --> 00:05:58.693
of one of the descendants of Rurik.

00:05:58.693 --> 00:06:01.076
So do you think it was actually this way?

00:06:01.076 --> 00:06:04.101
Or do you think the Varangians
maybe forced themselves

00:06:04.101 --> 00:06:07.796
on the Eastern Slavs and
later created this narrative,

00:06:07.796 --> 00:06:10.248
that they were invited to come in?

00:06:10.248 --> 00:06:12.062
But according to the Primary Chronicle,

00:06:12.062 --> 00:06:15.714
we have Rurik coming from Scandinavia

00:06:15.714 --> 00:06:19.547
to Novgorod and establishing
the land of Rus'.

00:06:21.008 --> 00:06:23.464
Now the word Rus is really interesting.

00:06:23.464 --> 00:06:26.254
Most historians believe
it to be the source

00:06:26.254 --> 00:06:28.467
of what we now say Russia,

00:06:28.467 --> 00:06:31.730
or even Belarus, which means white Rus.

00:06:31.730 --> 00:06:34.727
Some historians think
it comes from the name

00:06:34.727 --> 00:06:37.220
of Sweden at the time.

00:06:37.220 --> 00:06:41.387
Some believe that the Rus
were a subgroup of Varangians,

00:06:42.421 --> 00:06:43.517
of Vikings.

00:06:43.517 --> 00:06:45.872
Some believe that the word is derived

00:06:45.872 --> 00:06:47.903
from those who row.

00:06:47.903 --> 00:06:52.661
But either way, the Primary
Chronicle goes on to tell us

00:06:52.661 --> 00:06:55.411
from 870 to 879, on his deathbed,

00:06:56.627 --> 00:06:59.302
Rurik bequeathed his realm to Oleg,

00:06:59.302 --> 00:07:01.173
who belonged to his kin,

00:07:01.173 --> 00:07:04.841
and entrusted to Oleg's
hands his son Igor,

00:07:04.841 --> 00:07:07.373
for he was very young.

00:07:07.373 --> 00:07:11.626
And then from 880 to
882, Oleg set himself up

00:07:11.626 --> 00:07:15.626
as prince in Kiev and declared
that it should be the mother

00:07:15.626 --> 00:07:17.126
of Russian cities.

00:07:18.104 --> 00:07:21.354
So Rurik's immediate successor is Oleg.

00:07:22.383 --> 00:07:24.348
And in the early 880s,

00:07:24.348 --> 00:07:27.750
he goes and establishes himself in Kiev,

00:07:27.750 --> 00:07:30.000
expanding the land of Rus'.

00:07:31.332 --> 00:07:33.389
This is Kiev right over here,

00:07:33.389 --> 00:07:36.639
and because Oleg was able to take Kiev,

00:07:39.007 --> 00:07:42.168
the state that emerges from Rurik and Oleg

00:07:42.168 --> 00:07:44.615
not only is it known as
the Land of the Rus',

00:07:44.615 --> 00:07:47.162
but it's also known as the Kievan State,

00:07:47.162 --> 00:07:50.153
and they're often known
as the Kievan Rus'.

00:07:50.153 --> 00:07:52.885
And you can see here
how that state expands

00:07:52.885 --> 00:07:55.777
over the next few hundred years.

00:07:55.777 --> 00:07:57.870
As we get to the year 900,

00:07:57.870 --> 00:07:59.858
you have this off-white color,

00:07:59.858 --> 00:08:01.287
and you can see, it is in control

00:08:01.287 --> 00:08:03.454
of both Novgorod and Kiev.

00:08:04.317 --> 00:08:08.553
As you get to 1015, it's
taken even more territory,

00:08:08.553 --> 00:08:12.720
and by 1113, which is near
the peak of the Kievan state,

00:08:13.611 --> 00:08:14.858
you see that it has taken control

00:08:14.858 --> 00:08:18.306
of a good chunk of Eastern Europe.

00:08:18.306 --> 00:08:20.274
And as the state expands,

00:08:20.274 --> 00:08:22.731
its character changes as well.

00:08:22.731 --> 00:08:25.400
As you get to the end of the 10th century,

00:08:25.400 --> 00:08:29.420
you have a major event in
one of Rurik's descendants,

00:08:29.420 --> 00:08:33.045
Vladimir, often known
as Vladimir the Great,

00:08:33.045 --> 00:08:37.978
he decides to convert to
Eastern Orthodox Christianity.

00:08:37.978 --> 00:08:41.206
And in a future video, I might
talk about his rationales,

00:08:41.206 --> 00:08:45.162
or what historians view as
his rationales for conversion.

00:08:45.162 --> 00:08:47.508
And as we will see, over time,

00:08:47.508 --> 00:08:49.905
and because of not only his conversion,

00:08:49.905 --> 00:08:51.353
but essentially the conversion

00:08:51.353 --> 00:08:53.833
of the entire Kievan state,

00:08:53.833 --> 00:08:56.654
over time, especially
with the eventual decline

00:08:56.654 --> 00:08:58.934
of the Byzantine Empire,

00:08:58.934 --> 00:09:01.298
what would eventually be Russia

00:09:01.298 --> 00:09:06.257
becomes a center of Eastern
Orthodox Christianity.

00:09:06.257 --> 00:09:09.657
Now the Kievan state lasts
as an independent state

00:09:09.657 --> 00:09:12.957
until we get to the 13th century.

00:09:12.957 --> 00:09:14.315
And from many other videos,

00:09:14.315 --> 00:09:18.057
you might be guessing what
happens in the 13th century.

00:09:18.057 --> 00:09:20.883
You have Genghis Khan
and then his descendants

00:09:20.883 --> 00:09:24.347
emerge out of Central and Eastern Asia,

00:09:24.347 --> 00:09:27.952
and in 1240, you have the Mongol invasion,

00:09:27.952 --> 00:09:30.673
at which point, many of the principalities

00:09:30.673 --> 00:09:34.532
within the Land of Rus' become tributaries

00:09:34.532 --> 00:09:36.434
to the Mongolians.

00:09:36.434 --> 00:09:40.308
And they would be so for
the next, roughly, 200 years

00:09:40.308 --> 00:09:43.267
until Ivan the Great comes along

00:09:43.267 --> 00:09:47.453
and is able to exert
independence from the Mongols

00:09:47.453 --> 00:09:50.269
for the Rus, but we will cover that

00:09:50.269 --> 00:09:51.769
in a future video.


Types of discontinuities

WEBVTT
Kind: captions
Language: en

00:00:00.333 --> 00:00:01.795
- [Instructor] What we're
going to do in this video

00:00:01.795 --> 00:00:04.447
is talk about the various
types of discontinuities

00:00:04.447 --> 00:00:07.359
that you've probably seen
when you took algebra,

00:00:07.359 --> 00:00:11.010
or precalculus, but then
relate it to our understanding

00:00:11.010 --> 00:00:14.876
of both two-sided limits
and one-sided limits.

00:00:14.876 --> 00:00:18.727
So let's first review the
classification of discontinuities.

00:00:18.727 --> 00:00:22.274
So here on the left,
you see that this curve

00:00:22.274 --> 00:00:25.642
looks just like y equals x squared,

00:00:25.642 --> 00:00:28.502
until we get to x equals three.

00:00:28.502 --> 00:00:31.243
And instead of it being three squared,

00:00:31.243 --> 00:00:33.111
at this point you have this opening,

00:00:33.111 --> 00:00:35.893
and instead the function at
three is defined at four.

00:00:35.893 --> 00:00:37.420
But then it keeps going
and it looks just like

00:00:37.420 --> 00:00:39.543
y equals x squared.

00:00:39.543 --> 00:00:42.163
This is known as a point,

00:00:42.163 --> 00:00:44.663
or a removable, discontinuity.

00:00:45.834 --> 00:00:47.523
And it's called that for obvious reasons.

00:00:47.523 --> 00:00:49.821
You're discontinuous at that point.

00:00:49.821 --> 00:00:52.665
You might imagine defining
or redefining the function

00:00:52.665 --> 00:00:54.747
at that point so it is continuous,

00:00:54.747 --> 00:00:57.853
so that this discontinuity is removable.

00:00:57.853 --> 00:01:00.140
But then how does this
relate to our definition

00:01:00.140 --> 00:01:01.833
of continuity?

00:01:01.833 --> 00:01:05.243
Well, let's remind ourselves
our definition of continuity.

00:01:05.243 --> 00:01:07.772
We say f is continuous,

00:01:07.772 --> 00:01:08.689
continuous,

00:01:10.156 --> 00:01:11.406
if and only if,

00:01:12.266 --> 00:01:14.343
or let me write f continuous

00:01:14.343 --> 00:01:17.010
at x equals c, if and only if

00:01:18.094 --> 00:01:20.594
the limit as x approaches c

00:01:21.750 --> 00:01:26.565
of f of x is equal to the
actual value of the function

00:01:26.565 --> 00:01:28.739
when x is equal to c.

00:01:28.739 --> 00:01:30.714
So why does this one fail?

00:01:30.714 --> 00:01:33.460
Well, the two-sided limit actually exists.

00:01:33.460 --> 00:01:37.232
You could find, if we say
c in this case is three,

00:01:37.232 --> 00:01:38.708
the limit

00:01:38.708 --> 00:01:40.708
as x approaches three

00:01:41.637 --> 00:01:42.470
of f of x,

00:01:43.703 --> 00:01:46.412
it looks like, and if you
graphically inspect this,

00:01:46.412 --> 00:01:48.679
and I actually know this is the
graph of y equals x squared,

00:01:48.679 --> 00:01:51.410
except at that discontinuity
right over there,

00:01:51.410 --> 00:01:54.066
this is equal to nine.

00:01:54.066 --> 00:01:57.511
But the issue is, the way
this graph has been depicted,

00:01:57.511 --> 00:02:00.342
this is not the same thing
as the value of the function.

00:02:00.342 --> 00:02:01.909
This function

00:02:01.909 --> 00:02:04.864
f of three, the way it's been graphed,

00:02:04.864 --> 00:02:07.890
f of three is equal to four.

00:02:07.890 --> 00:02:11.305
So this is a situation where
this two-sided limit exists,

00:02:11.305 --> 00:02:14.679
but it's not equal to the
value of that function.

00:02:14.679 --> 00:02:16.594
You might see other
circumstances where the function

00:02:16.594 --> 00:02:18.144
isn't even defined there,

00:02:18.144 --> 00:02:20.144
so that isn't even there.

00:02:20.144 --> 00:02:22.391
And so, once again, the limit might exist,

00:02:22.391 --> 00:02:24.437
but the function might
not be defined there.

00:02:24.437 --> 00:02:28.273
So, in either case, you aren't
going to meet this criteria

00:02:28.273 --> 00:02:29.523
for continuity.

00:02:30.427 --> 00:02:34.153
And so that's how a point
or removable discontinuity,

00:02:34.153 --> 00:02:36.169
why it is discontinuous

00:02:36.169 --> 00:02:40.770
with regards to our limit
definition of continuity.

00:02:40.770 --> 00:02:43.281
So now let's look at this second example.

00:02:43.281 --> 00:02:45.924
If we looked at our
intuitive continuity test,

00:02:45.924 --> 00:02:48.629
if we would just try to trace this thing,

00:02:48.629 --> 00:02:52.461
we see that once we get to x equals two,

00:02:52.461 --> 00:02:55.139
I have to pick up my
pencil to keep tracing it.

00:02:55.139 --> 00:02:58.222
And so that's a pretty good
sign that we are discontinuous.

00:02:58.222 --> 00:03:00.512
We see that over here as well.

00:03:00.512 --> 00:03:03.595
If I'm tracing this function,
I gotta pick up my pencil to,

00:03:03.595 --> 00:03:04.518
I can't go to that point.

00:03:04.518 --> 00:03:06.018
I have to jump down here,

00:03:06.018 --> 00:03:07.681
and then keep going right over there.

00:03:07.681 --> 00:03:09.686
So in either case I have
to pick up my pencil.

00:03:09.686 --> 00:03:12.355
And so, intuitively, it is discontinuous.

00:03:12.355 --> 00:03:14.934
But this particular type of discontinuity,

00:03:14.934 --> 00:03:17.381
where I am making a jump from one point,

00:03:17.381 --> 00:03:19.584
and then I'm making a jump
down here to continue,

00:03:19.584 --> 00:03:22.379
it is intuitively called a jump

00:03:22.379 --> 00:03:23.546
discontinuity,

00:03:24.432 --> 00:03:25.599
discontinuity.

00:03:27.754 --> 00:03:31.245
And this is, of course, a
point removable discontinuity.

00:03:31.245 --> 00:03:33.775
And so how does this relate to limits?

00:03:33.775 --> 00:03:37.704
Well, here, the left and
right-handed limits exist,

00:03:37.704 --> 00:03:39.242
but they're not the same thing,

00:03:39.242 --> 00:03:41.925
so you don't have a two-sided limit.

00:03:41.925 --> 00:03:45.566
So, for example, for
this one in particular,

00:03:45.566 --> 00:03:48.580
for all the x-values up to
and including x equals two,

00:03:48.580 --> 00:03:51.022
this is the graph of y equals x squared.

00:03:51.022 --> 00:03:53.159
And then for x greater than two,

00:03:53.159 --> 00:03:55.179
it's the graph of square root of x.

00:03:55.179 --> 00:03:57.059
So in this scenario,

00:03:57.059 --> 00:03:59.417
if you were to take the limit

00:03:59.417 --> 00:04:00.250
of f of x

00:04:01.502 --> 00:04:03.002
as x approaches

00:04:04.209 --> 00:04:05.042
two

00:04:06.000 --> 00:04:07.167
from the left,

00:04:08.191 --> 00:04:09.570
from the left,

00:04:09.570 --> 00:04:11.010
this is going to be equal to four,

00:04:11.010 --> 00:04:12.192
you're approaching this value.

00:04:12.192 --> 00:04:14.683
And that actually is the
value of the function.

00:04:14.683 --> 00:04:18.598
But if you were to take the
limit as x approaches two

00:04:18.598 --> 00:04:20.995
from the right of f of x,

00:04:20.995 --> 00:04:22.881
what is that going to be equal to?

00:04:22.881 --> 00:04:24.070
Well, approaching from the right,

00:04:24.070 --> 00:04:25.534
this is actually the square root of x,

00:04:25.534 --> 00:04:28.606
so it's approaching
the square root of two.

00:04:28.606 --> 00:04:29.714
You wouldn't know it's
the square root of two

00:04:29.714 --> 00:04:30.716
just by looking at this.

00:04:30.716 --> 00:04:32.417
I know that, just because when I,

00:04:32.417 --> 00:04:34.394
when I went on to Desmos
and defined the function,

00:04:34.394 --> 00:04:36.157
that's the function that I used.

00:04:36.157 --> 00:04:37.842
But it's clear even visually

00:04:37.842 --> 00:04:39.586
that you're approaching
two different values

00:04:39.586 --> 00:04:41.066
when you approach from the left

00:04:41.066 --> 00:04:42.770
than when you approach from the right.

00:04:42.770 --> 00:04:44.917
So even though the one-sided limits exist,

00:04:44.917 --> 00:04:46.401
they're not approaching the same thing,

00:04:46.401 --> 00:04:48.230
so the two-sided limit doesn't exist.

00:04:48.230 --> 00:04:49.850
And if the two-sided limit doesn't exist,

00:04:49.850 --> 00:04:51.541
it for sure cannot be equal to the value

00:04:51.541 --> 00:04:54.508
of the function there, even
if the function is defined.

00:04:54.508 --> 00:04:58.744
So that's why the jump
discontinuity is failing this test.

00:04:58.744 --> 00:04:59.885
Now, once again, it's intuitive.

00:04:59.885 --> 00:05:01.459
You're seeing that, hey, I gotta jump,

00:05:01.459 --> 00:05:02.546
I gotta pick up my pencil.

00:05:02.546 --> 00:05:06.158
These two things are not
connected to each other.

00:05:06.158 --> 00:05:08.752
Finally, what you see here is,

00:05:08.752 --> 00:05:10.000
when you learned precalculus,

00:05:10.000 --> 00:05:13.617
often known as an
asymptotic discontinuity,

00:05:13.617 --> 00:05:14.534
asymptotic,

00:05:17.462 --> 00:05:19.124
asymptotic

00:05:19.124 --> 00:05:20.291
discontinuity,

00:05:21.508 --> 00:05:22.675
discontinuity.

00:05:23.780 --> 00:05:27.525
And, intuitively, you
have an asymptote here.

00:05:27.525 --> 00:05:30.388
It's a vertical asymptote at x equals two.

00:05:30.388 --> 00:05:33.602
If I were to try to trace the graph

00:05:33.602 --> 00:05:34.855
from the left,

00:05:34.855 --> 00:05:36.803
I would just keep on going.

00:05:36.803 --> 00:05:40.134
In fact, I would be doing
it forever, 'cause it's,

00:05:40.134 --> 00:05:42.126
it would be infinitely,

00:05:42.126 --> 00:05:44.484
it would be unbounded as
I get closer and closer

00:05:44.484 --> 00:05:46.332
to x equals two from the left.

00:05:46.332 --> 00:05:48.936
And if try to get to x
equals two from the right,

00:05:48.936 --> 00:05:51.132
once again I get unbounded up.

00:05:51.132 --> 00:05:52.757
But even if I could,

00:05:52.757 --> 00:05:55.067
and when I say it's unbounded,
it goes to infinity,

00:05:55.067 --> 00:05:57.317
so it's actually impossible

00:05:58.634 --> 00:06:02.367
in a mortal's lifespan to
try to trace the whole thing.

00:06:02.367 --> 00:06:04.418
But you get the sense that,
hey, there's no way that I could

00:06:04.418 --> 00:06:08.656
draw from here to here
without picking up my pencil.

00:06:08.656 --> 00:06:12.466
And if you wanna relate it
to our notion of limits,

00:06:12.466 --> 00:06:13.715
it's that

00:06:13.715 --> 00:06:16.930
both the left and right-handed
limits are unbounded,

00:06:16.930 --> 00:06:18.398
so they officially don't exist.

00:06:18.398 --> 00:06:21.675
So if they don't exist, then
we can't meet these conditions.

00:06:21.675 --> 00:06:23.076
So if I were to say,

00:06:23.076 --> 00:06:24.363
the limit

00:06:24.363 --> 00:06:28.450
as x approaches two from the
left-hand side of f of x,

00:06:28.450 --> 00:06:31.053
we can see that it goes unbounded
in the negative direction.

00:06:31.053 --> 00:06:33.352
You might sometimes see someone
write something like this,

00:06:33.352 --> 00:06:34.601
negative infinity.

00:06:34.601 --> 00:06:36.975
But that's a little
handwavy with the math.

00:06:36.975 --> 00:06:41.010
The more correct way to say
it is it's just unbounded,

00:06:41.010 --> 00:06:42.617
unbounded.

00:06:42.617 --> 00:06:44.920
And, likewise, if we
thought about the limit

00:06:44.920 --> 00:06:46.751
as x approaches two

00:06:46.751 --> 00:06:48.606
from the right

00:06:48.606 --> 00:06:49.918
of f of x,

00:06:49.918 --> 00:06:52.953
it is now unbounded
towards positive infinity.

00:06:52.953 --> 00:06:54.367
So this, once again,

00:06:54.367 --> 00:06:55.786
this is also,

00:06:55.786 --> 00:06:57.983
this is also unbounded.

00:06:57.983 --> 00:06:59.297
And

00:06:59.297 --> 00:07:01.440
because it's unbounded and
this limit does not exist,

00:07:01.440 --> 00:07:02.631
it can't meet these conditions.

00:07:02.631 --> 00:07:04.950
And so we are going to be discontinuous.

00:07:04.950 --> 00:07:07.696
So this is a point or
removable discontinuity,

00:07:07.696 --> 00:07:09.931
jump discontinuity, I'm jumping,

00:07:09.931 --> 00:07:12.217
and then we have these
asymptotes, a vertical asymptote.

00:07:12.217 --> 00:07:15.045
This is an asymptotic discontinuity.


Functions with same limit at infinity

WEBVTT
Kind: captions
Language: en

00:00:00.465 --> 00:00:01.723
- [Instructor] The goal of this video

00:00:01.723 --> 00:00:04.526
is to get an appreciation
that you could have many,

00:00:04.526 --> 00:00:06.231
in fact, you could have an infinite number

00:00:06.231 --> 00:00:09.080
of functions that have the same limit

00:00:09.080 --> 00:00:11.317
as x approaches infinity.

00:00:11.317 --> 00:00:12.945
So, if we were to make
the general statement

00:00:12.945 --> 00:00:15.016
that the limit of some function f of x,

00:00:15.016 --> 00:00:17.750
as x approaches infinity,
is equal to three.

00:00:17.750 --> 00:00:18.805
What I wanna do in this video

00:00:18.805 --> 00:00:20.801
is show some examples of that.

00:00:20.801 --> 00:00:22.261
And to show that we can keep creating

00:00:22.261 --> 00:00:24.290
more and more examples,
really an infinite number

00:00:24.290 --> 00:00:26.813
of examples where that
is going to be true.

00:00:26.813 --> 00:00:30.987
So, for example, we could
look at this graph over here.

00:00:30.987 --> 00:00:34.000
And in other videos, we'll think
about why this is the case,

00:00:34.000 --> 00:00:35.179
but just think about what happens when

00:00:35.179 --> 00:00:37.376
you have very, very large Xs.

00:00:37.376 --> 00:00:38.960
When you have very, very large Xs,

00:00:38.960 --> 00:00:40.825
the plus five doesn't matter as much,

00:00:40.825 --> 00:00:43.262
and so it gets closer and
closer to three x squared

00:00:43.262 --> 00:00:45.355
over x squared, which is equal to three.

00:00:45.355 --> 00:00:47.025
And you could see that right over here,

00:00:47.025 --> 00:00:49.176
it's graphed in this green color.

00:00:49.176 --> 00:00:51.856
And you can see, even
when x is equal to 10,

00:00:51.856 --> 00:00:54.374
we're getting awfully close
to three right over there.

00:00:54.374 --> 00:00:56.561
Let me zoom out a little
bit so you see our axes.

00:00:56.561 --> 00:00:57.710
So that is three.

00:00:57.710 --> 00:01:01.170
Let me draw a dotted
line at the asymptote.

00:01:01.170 --> 00:01:04.217
That is y is equal to
three, and so you see

00:01:04.217 --> 00:01:05.802
the function's getting closer and closer

00:01:05.802 --> 00:01:07.387
as x approaches infinity.

00:01:07.387 --> 00:01:09.256
But that's not the only
function that could do that,

00:01:09.256 --> 00:01:10.682
as I keep saying, there's
an infinite number

00:01:10.682 --> 00:01:11.940
of functions that could do that.

00:01:11.940 --> 00:01:14.015
You could have this somewhat wild function

00:01:14.015 --> 00:01:15.764
that involves natural logs.

00:01:15.764 --> 00:01:18.772
That too, as x approaches infinity,

00:01:18.772 --> 00:01:22.142
it is getting closer and closer to three.

00:01:22.142 --> 00:01:23.404
It might be getting closer to three

00:01:23.404 --> 00:01:26.168
at a slightly slower rate
than the one in green,

00:01:26.168 --> 00:01:27.551
but we're talking about infinity.

00:01:27.551 --> 00:01:30.676
As x approaches infinity, this
thing is approaching three.

00:01:30.676 --> 00:01:32.504
And as we've talked about in other videos,

00:01:32.504 --> 00:01:34.089
you could even have things that keep

00:01:34.089 --> 00:01:35.719
oscillating around the asymptote,

00:01:35.719 --> 00:01:39.527
as long as they're getting
closer and closer and closer

00:01:39.527 --> 00:01:43.286
to it as x gets larger
and larger and larger.

00:01:43.286 --> 00:01:45.679
So, for example, that
function right over there.

00:01:45.679 --> 00:01:46.817
Let me zoom in.

00:01:46.817 --> 00:01:48.317
So, let's zoom in.

00:01:50.152 --> 00:01:53.196
Let's say when x is equal to 14,

00:01:53.196 --> 00:01:55.226
we can see that they're
all approaching three.

00:01:55.226 --> 00:01:57.297
The purple one is oscillating around it,

00:01:57.297 --> 00:02:00.706
the other two are
approaching three from below.

00:02:00.706 --> 00:02:02.573
But as we get much larger,

00:02:02.573 --> 00:02:05.626
let me actually zoom out a
ways, and then I'll zoom in.

00:02:05.626 --> 00:02:09.122
So let's get to really large values.

00:02:09.122 --> 00:02:12.057
So, actually, even 100
isn't even that large

00:02:12.057 --> 00:02:13.968
if we're thinking about infinity.

00:02:13.968 --> 00:02:15.348
Even a trillion wouldn't be that large

00:02:15.348 --> 00:02:16.401
if we're thinking about infinity.

00:02:16.401 --> 00:02:17.986
But let's go to 200.

00:02:17.986 --> 00:02:20.954
200 is much larger than
numbers we've been looking at.

00:02:20.954 --> 00:02:22.498
And let me zoom in

00:02:22.498 --> 00:02:25.380
when x is equal to 200, and you can see,

00:02:25.380 --> 00:02:29.770
we have to zoom in an
awfully lot, an awful lot,

00:02:29.770 --> 00:02:34.161
just to even see that the
graphs still aren't quite

00:02:34.161 --> 00:02:36.235
stabilized around the asymptote,

00:02:36.235 --> 00:02:39.450
that they are a little bit
different than the asymptote.

00:02:39.450 --> 00:02:41.358
I really zoomed in, I
mean look at the scale.

00:02:41.358 --> 00:02:45.094
This is, each of these are
now 100th, each square.

00:02:45.094 --> 00:02:46.147
And so we've gotten much, much,

00:02:46.147 --> 00:02:47.975
much closer to the asymptote.

00:02:47.975 --> 00:02:49.156
In fact, the green function,

00:02:49.156 --> 00:02:50.659
we still can't tell the difference.

00:02:50.659 --> 00:02:53.054
You can see the calculation, this is up to

00:02:53.054 --> 00:02:54.517
three or four decimal places,

00:02:54.517 --> 00:02:56.955
we're getting awfully close to three now,

00:02:56.955 --> 00:02:58.419
but we aren't there.

00:02:58.419 --> 00:03:00.975
So the green functions
got there the fastest,

00:03:00.975 --> 00:03:02.315
is an argument.

00:03:02.315 --> 00:03:05.729
But the whole point of this
is to emphasize the fact

00:03:05.729 --> 00:03:08.493
that there's an infinite
number of functions

00:03:08.493 --> 00:03:11.171
for which you could make
the statement that we made.

00:03:11.171 --> 00:03:15.480
That the limit of the function
as x approaches infinity,

00:03:15.480 --> 00:03:17.875
in this case, we said that limit is going

00:03:17.875 --> 00:03:21.814
to be equal to three, and I
just picked three arbitrarily.

00:03:21.814 --> 00:03:26.079
This could be true for
any, for any function.

00:03:26.079 --> 00:03:28.764
I didn't realize how much I had zoomed in.

00:03:28.764 --> 00:03:31.488
So let me now go back to the origin

00:03:31.488 --> 00:03:34.411
where we had our original expression.

00:03:34.411 --> 00:03:37.947
So, there we have it, and
maybe I can zoom in this way.

00:03:37.947 --> 00:03:39.533
So there you have it.

00:03:39.533 --> 00:03:42.416
Limit of any of these,
as x approaches infinity,

00:03:42.416 --> 00:03:43.916
is equal to three.


Infinite limits and asymptotes

WEBVTT
Kind: captions
Language: en

00:00:00.981 --> 00:00:02.122
- [Instructor] What we're
going to do in this video

00:00:02.122 --> 00:00:05.380
is use the online graphing
calculator Desmos,

00:00:05.380 --> 00:00:07.957
and explore the relationship
between vertical

00:00:07.957 --> 00:00:10.680
and horizontal asymptotes, and think about

00:00:10.680 --> 00:00:12.601
how they relate to what we know

00:00:12.601 --> 00:00:14.189
about limits.

00:00:14.189 --> 00:00:18.136
So let's first graph two over x minus one,

00:00:18.136 --> 00:00:20.406
so let me get that one graphed,

00:00:20.406 --> 00:00:22.960
and so you can immediately
see that something interesting

00:00:22.960 --> 00:00:25.627
happens at x is equal to one.

00:00:27.001 --> 00:00:29.237
If you were to just
substitute x equals one

00:00:29.237 --> 00:00:31.367
into this expression, you're going to get

00:00:31.367 --> 00:00:35.986
two over zero, and whenever
you get a non-zero thing,

00:00:35.986 --> 00:00:39.496
over zero, that's a
good sign that you might

00:00:39.496 --> 00:00:43.202
be dealing with a vertical asymptote.

00:00:43.202 --> 00:00:45.489
In fact we can draw
that vertical asymptote

00:00:45.489 --> 00:00:48.099
right over here at x equals one.

00:00:48.099 --> 00:00:50.721
But let's think about how
that relates to limits.

00:00:50.721 --> 00:00:53.688
What if we were to explore
the limit as x approaches one

00:00:53.688 --> 00:00:57.432
of f of x is equal to
two over x minus one,

00:00:57.432 --> 00:00:59.168
and we could think about it from the left

00:00:59.168 --> 00:01:01.061
and from the right, so if we approach one

00:01:01.061 --> 00:01:03.358
from the left, let me zoom in a little bit

00:01:03.358 --> 00:01:06.775
over here, so we can see as
we approach from the left

00:01:06.775 --> 00:01:09.483
when x is equal to zero, the f of x

00:01:09.483 --> 00:01:12.093
would be equal to negative two,

00:01:12.093 --> 00:01:15.042
when x is equal to point five, f of x

00:01:15.042 --> 00:01:17.173
is equal to negative of
four, and then it just gets

00:01:17.173 --> 00:01:19.459
more and more negative the closer we get

00:01:19.459 --> 00:01:21.958
to one from the left.

00:01:21.958 --> 00:01:25.694
I could really, so I'm
not even that close yet

00:01:25.694 --> 00:01:29.861
if I get to let's say 0.91,
I'm still nine hundredths

00:01:31.892 --> 00:01:36.539
less than one, I'm at
negative 22.222, already.

00:01:36.539 --> 00:01:39.910
And so the limit as we
approach one from the left

00:01:39.910 --> 00:01:42.199
is unbounded, some people would say

00:01:42.199 --> 00:01:43.794
it goes to negative
infinity, but it's really

00:01:43.794 --> 00:01:45.925
an undefined limit, it is unbounded

00:01:45.925 --> 00:01:47.673
in the negative direction.

00:01:47.673 --> 00:01:50.779
And likewise, as we
approach from the right,

00:01:50.779 --> 00:01:55.103
we get unbounded in the
positive infinity direction

00:01:55.103 --> 00:01:57.905
and technically we would
say that that limit

00:01:57.905 --> 00:01:59.904
does not exist.

00:01:59.904 --> 00:02:01.972
And this would be the
case when we're dealing

00:02:01.972 --> 00:02:06.006
with a vertical asymptote
like we see over here.

00:02:06.006 --> 00:02:08.705
Now let's compare that
to a horizontal asymptote

00:02:08.705 --> 00:02:11.823
where it turns out that the limit

00:02:11.823 --> 00:02:14.013
actually can exist.

00:02:14.013 --> 00:02:17.930
So let me delete these or
just erase them for now,

00:02:17.930 --> 00:02:19.130
and so let's look at this function

00:02:19.130 --> 00:02:20.956
which is a pretty neat
function, I made it up

00:02:20.956 --> 00:02:22.423
right before this video started

00:02:22.423 --> 00:02:25.423
but it's kind of cool
looking, but let's think

00:02:25.423 --> 00:02:29.964
about the behavior as
x approaches infinity.

00:02:29.964 --> 00:02:33.269
So as x approaches infinity,
it looks like our y value

00:02:33.269 --> 00:02:35.010
or the value of the
expression, if we said y

00:02:35.010 --> 00:02:37.416
is equal to that expression, it looks like

00:02:37.416 --> 00:02:40.680
it's getting closer and
closer and closer to three.

00:02:40.680 --> 00:02:44.174
And so we could say that we
have a horizontal asymptote

00:02:44.174 --> 00:02:47.193
at y is equal to three, and we could also

00:02:47.193 --> 00:02:49.304
and there's a more rigorous
way of defining it,

00:02:49.304 --> 00:02:52.581
say that our limit as
x approaches infinity

00:02:52.581 --> 00:02:55.650
is equal of the expression
or of the function,

00:02:55.650 --> 00:02:57.150
is equal to three.

00:02:58.152 --> 00:03:00.171
Notice my mouse is
covering it a little bit

00:03:00.171 --> 00:03:02.495
as we get larger and larger, we're getting

00:03:02.495 --> 00:03:03.948
closer and closer to three,

00:03:03.948 --> 00:03:06.082
in fact we're getting
so close now, well here

00:03:06.082 --> 00:03:08.571
you can see we're
getting closer and closer

00:03:08.571 --> 00:03:10.238
and closer to three.

00:03:11.866 --> 00:03:13.317
And you could also
think about what happens

00:03:13.317 --> 00:03:15.469
as x approaches negative infinity and here

00:03:15.469 --> 00:03:17.077
you're getting closer
and closer and closer

00:03:17.077 --> 00:03:18.649
to three from below.

00:03:18.649 --> 00:03:20.845
Now one thing that's
interesting about horizontal

00:03:20.845 --> 00:03:23.283
asymptotes is you might
see that the function

00:03:23.283 --> 00:03:25.692
actually can cross a horizontal asymptote.

00:03:25.692 --> 00:03:27.662
It's crossing this horizontal asymptote

00:03:27.662 --> 00:03:30.537
in this area in between and
even as we approach infinity

00:03:30.537 --> 00:03:32.639
or negative infinity, you can oscillate

00:03:32.639 --> 00:03:34.621
around that horizontal asymptote.

00:03:34.621 --> 00:03:38.788
Let me set this up, let me
multiply this times sine of x.

00:03:40.297 --> 00:03:41.933
And so there you have it,
we are now oscillating

00:03:41.933 --> 00:03:44.600
around the horizontal asymptote,

00:03:45.718 --> 00:03:47.116
and once again this limit can exist

00:03:47.116 --> 00:03:49.752
even though we keep crossing
the horizontal asymptote,

00:03:49.752 --> 00:03:52.108
we're getting closer and
closer and closer to it

00:03:52.108 --> 00:03:53.858
the larger x gets.

00:03:55.047 --> 00:03:56.727
And that's actually the
key difference between

00:03:56.727 --> 00:03:58.178
a horizontal and a vertical asymptote.

00:03:58.178 --> 00:04:00.336
Vertical asymptotes if you're
dealing with a function,

00:04:00.336 --> 00:04:03.122
you're not going to cross
it, while with a horizontal

00:04:03.122 --> 00:04:05.676
asymptote, you could,
and you are just getting

00:04:05.676 --> 00:04:07.102
closer and closer and closer to it

00:04:07.102 --> 00:04:11.262
as x goes to positive infinity or as x

00:04:11.262 --> 00:04:13.024
goes to negative infinity.