WEBVTT

00:00:00.000 --> 00:00:07.300
PROFESSOR: Hi.

00:00:07.300 --> 00:00:08.850
Welcome back to recitation.

00:00:08.850 --> 00:00:10.920
You've been talking
about computing limits

00:00:10.920 --> 00:00:12.580
of some indeterminate forms.

00:00:12.580 --> 00:00:15.260
In particular, you
used l'Hopital's rule

00:00:15.260 --> 00:00:18.940
to help you out with some
limits in the form 0 over 0

00:00:18.940 --> 00:00:20.540
or infinity over infinity.

00:00:20.540 --> 00:00:22.890
So these are the two
indeterminate ratios.

00:00:22.890 --> 00:00:26.760
And you also--
so, OK so when you

00:00:26.760 --> 00:00:29.404
have a limit that is in the
form of an indeterminate ratio,

00:00:29.404 --> 00:00:30.820
you've seen that
one tool that you

00:00:30.820 --> 00:00:34.080
can use to help compute the
limit is l'Hopital's rule.

00:00:34.080 --> 00:00:38.050
There are other indeterminate
forms for limits as well.

00:00:38.050 --> 00:00:40.980
So you actually saw,
in lecture, another one

00:00:40.980 --> 00:00:45.080
of these, which was you saw a
limit of the form 0 to the 0.

00:00:45.080 --> 00:00:48.290
So that was the limit
you saw, was the limit

00:00:48.290 --> 00:00:52.360
as x goes to 0 from the
right of x to the x.

00:00:52.360 --> 00:00:54.370
So that was going to 0 over 0.

00:00:54.370 --> 00:00:57.930
And the two competing forces are
that as the base of the limit

00:00:57.930 --> 00:01:00.940
goes to 0, that wants to make
the whole thing get closer

00:01:00.940 --> 00:01:01.830
to 0.

00:01:01.830 --> 00:01:03.650
And when the exponent
is going to 0,

00:01:03.650 --> 00:01:06.080
that makes the whole thing
want to get closer to 1.

00:01:06.080 --> 00:01:07.846
So you have those
two competing forces.

00:01:07.846 --> 00:01:09.470
That's why it is an
indeterminate form.

00:01:09.470 --> 00:01:10.995
When you were
solving this limit,

00:01:10.995 --> 00:01:15.780
you-- first thing you did was
you wrote it as an exponential

00:01:15.780 --> 00:01:16.790
in base e.

00:01:16.790 --> 00:01:21.410
So you wrote x to the
x as e to the x*ln x.

00:01:21.410 --> 00:01:24.470
So x*ln x is also an
indeterminate form as x goes

00:01:24.470 --> 00:01:25.280
to 0.

00:01:25.280 --> 00:01:30.490
It's an indeterminate form
of the form 0 times infinity.

00:01:30.490 --> 00:01:32.180
Here I'm writing
infinity to mean

00:01:32.180 --> 00:01:34.910
either positive infinity
or negative infinity.

00:01:34.910 --> 00:01:37.510
So in this case, this
is an indeterminate form

00:01:37.510 --> 00:01:40.000
because when you have
one factor going to 0,

00:01:40.000 --> 00:01:42.406
that makes the whole product
want to get closer to 0.

00:01:42.406 --> 00:01:44.530
Whereas when you have one
factor going to infinity,

00:01:44.530 --> 00:01:46.904
either positive or negative,
that makes the whole product

00:01:46.904 --> 00:01:47.850
want to get big.

00:01:47.850 --> 00:01:50.140
So that's why it
becomes indeterminate.

00:01:50.140 --> 00:01:54.320
And you were able to evaluate
the limit x*ln x, in that case,

00:01:54.320 --> 00:01:57.500
by rewriting it as a quotient.

00:01:57.500 --> 00:02:00.000
There are a few other
indeterminate forms

00:02:00.000 --> 00:02:01.990
that I'd like to mention.

00:02:01.990 --> 00:02:05.810
So in particular, there are
three other indeterminate forms

00:02:05.810 --> 00:02:07.890
different from these.

00:02:07.890 --> 00:02:10.510
And then I'll give you
an example of one of them

00:02:10.510 --> 00:02:12.300
to solve a problem.

00:02:12.300 --> 00:02:15.420
so the three other
indeterminate forms,

00:02:15.420 --> 00:02:18.190
there are two more that are sort
of an exponential indeterminate

00:02:18.190 --> 00:02:21.100
form, and there's
one sort of outlier.

00:02:21.100 --> 00:02:25.400
So one of the other
indeterminate forms

00:02:25.400 --> 00:02:28.750
is infinity to the 0.

00:02:28.750 --> 00:02:32.440
So when you have an
exponential expression where

00:02:32.440 --> 00:02:34.780
the base is getting
very, very large,

00:02:34.780 --> 00:02:37.317
and the exponent is going to
0, well the base getting large

00:02:37.317 --> 00:02:38.775
makes the whole
fraction want to be

00:02:38.775 --> 00:02:42.720
sorry the whole expression
want to be large,

00:02:42.720 --> 00:02:45.445
whereas the exponent going to 0
makes the whole expression want

00:02:45.445 --> 00:02:46.706
to get closer to 1.

00:02:46.706 --> 00:02:48.330
And so those two
forces are in tension,

00:02:48.330 --> 00:02:49.970
and you can end up
with limits that

00:02:49.970 --> 00:02:53.920
equal any value when you
have a limit like this,

00:02:53.920 --> 00:02:55.580
infinity to the 0.

00:02:55.580 --> 00:03:00.410
Another similar one
is 1 to the infinity.

00:03:00.410 --> 00:03:03.250
And this is the one I'll give
you an exercise on in a minute.

00:03:03.250 --> 00:03:05.780
So when you have a
limit of the form 1

00:03:05.780 --> 00:03:08.820
to the infinity, so something to
the something where the base is

00:03:08.820 --> 00:03:11.230
going to 1 and the exponent
is going to infinity,

00:03:11.230 --> 00:03:14.480
the base going to 1 makes this
whole thing want to go to 1.

00:03:14.480 --> 00:03:17.420
The exponent to infinity
makes this whole thing

00:03:17.420 --> 00:03:20.890
want to either blow up if
it's a little bigger than 1,

00:03:20.890 --> 00:03:23.580
or get really small if it's
a little smaller than 1.

00:03:23.580 --> 00:03:25.580
So you have, again,
a tension there,

00:03:25.580 --> 00:03:27.730
and the result is indeterminate.

00:03:27.730 --> 00:03:31.400
The one sort of unusual
one that you have is also--

00:03:31.400 --> 00:03:33.540
which I'm not going to
talk much more about--

00:03:33.540 --> 00:03:35.460
is infinity minus infinity.

00:03:37.315 --> 00:03:38.940
So when you have two
very large things,

00:03:38.940 --> 00:03:41.840
so here I mean either
positive infinity

00:03:41.840 --> 00:03:44.280
minus positive infinity,
or negative infinity

00:03:44.280 --> 00:03:45.500
minus negative infinity.

00:03:45.500 --> 00:03:48.347
When you have two things that
are both getting very large,

00:03:48.347 --> 00:03:50.430
their difference could
also be getting very large,

00:03:50.430 --> 00:03:51.846
or it could be
getting very small,

00:03:51.846 --> 00:03:55.380
or could be doing
anything in between.

00:03:55.380 --> 00:03:57.820
So that's also an
indeterminate form.

00:03:57.820 --> 00:04:00.780
So these are the seven
indeterminate forms.

00:04:00.780 --> 00:04:03.600
When you have a
quotient, you can always

00:04:03.600 --> 00:04:04.920
apply l'Hopital's rule.

00:04:04.920 --> 00:04:06.810
When you have a
product, you can always

00:04:06.810 --> 00:04:08.730
rewrite it as a
quotient, by writing

00:04:08.730 --> 00:04:15.122
for example, x squared
plus 1 times-- well,

00:04:15.122 --> 00:04:22.010
let's see-- e to the minus x.

00:04:22.010 --> 00:04:23.110
That's a product.

00:04:23.110 --> 00:04:25.600
And you could always
rewrite it as a quotient,

00:04:25.600 --> 00:04:32.710
for example, e to the minus x
over 1 over x squared plus 1.

00:04:32.710 --> 00:04:34.100
There might even
be a smarter way

00:04:34.100 --> 00:04:35.810
to rewrite this
product as a quotient.

00:04:38.174 --> 00:04:38.674
But, OK.

00:04:38.674 --> 00:04:41.340
And so when you have an
exponential, as in this case,

00:04:41.340 --> 00:04:43.926
you can use this
rewriting in base e trick

00:04:43.926 --> 00:04:45.800
to turn it into a product,
which you can then

00:04:45.800 --> 00:04:48.209
turn into a quotient.

00:04:48.209 --> 00:04:50.500
For the difference case, the
reason I say it's unusual,

00:04:50.500 --> 00:04:54.380
is just that there's not
a good, general method

00:04:54.380 --> 00:04:55.710
for working with these.

00:04:55.710 --> 00:04:58.087
There are a lot
of special cases.

00:04:58.087 --> 00:04:59.670
And you sort of have
to or what I mean

00:04:59.670 --> 00:05:01.890
is you have to analyze them
on a case by case basis.

00:05:01.890 --> 00:05:06.120
There are different sort of
techniques that will work.

00:05:06.120 --> 00:05:09.350
So let me give you
an example of one

00:05:09.350 --> 00:05:11.949
of these 1 to the
infinity kinds.

00:05:11.949 --> 00:05:13.240
So why don't we come over here.

00:05:13.240 --> 00:05:16.520
So compute the
limit as x goes to 0

00:05:16.520 --> 00:05:21.170
from the right of 1 plus
3x to the 10 divided by x.

00:05:21.170 --> 00:05:25.180
So we see as x is going to
0, this base is going to 1.

00:05:25.180 --> 00:05:27.250
And so that makes this
whole expression want

00:05:27.250 --> 00:05:32.790
to be close to 1, whereas this
exponent is going to infinity,

00:05:32.790 --> 00:05:34.840
since this is just a
little bit bigger than 1,

00:05:34.840 --> 00:05:37.100
that makes the whole
expression want to be big

00:05:37.100 --> 00:05:40.090
when the exponent is big.

00:05:40.090 --> 00:05:43.950
So you have a tension here
between the base going to 1

00:05:43.950 --> 00:05:45.450
and the exponent
going to infinity.

00:05:45.450 --> 00:05:48.144
So the question is, try and
actually compute this limit.

00:05:48.144 --> 00:05:50.560
So why don't you pause the
video, take a couple of minutes

00:05:50.560 --> 00:05:52.180
to work on this
question, come back,

00:05:52.180 --> 00:05:53.471
and we can work on it together.

00:06:03.440 --> 00:06:04.050
Welcome back.

00:06:04.050 --> 00:06:07.250
Hopefully you had some luck
working on this problem.

00:06:07.250 --> 00:06:11.270
As we said, this is a limit of
an indeterminate form of the 1

00:06:11.270 --> 00:06:12.420
to infinity type.

00:06:12.420 --> 00:06:15.410
As one of the three exponential
types of indeterminate forms,

00:06:15.410 --> 00:06:19.070
a really promising first
step almost every time

00:06:19.070 --> 00:06:22.090
is to rewrite this as a
exponential expression

00:06:22.090 --> 00:06:23.650
with the base e.

00:06:23.650 --> 00:06:26.470
So we just do-- first we just
do an algebraic manipulation

00:06:26.470 --> 00:06:29.620
on the thing we're taking
the limit of, and then often,

00:06:29.620 --> 00:06:32.440
very often, that simplifies
it into something that we can

00:06:32.440 --> 00:06:34.040
actually compute the limit of.

00:06:34.040 --> 00:06:38.460
So in particular, we
have that 1 plus 3x,

00:06:38.460 --> 00:06:41.600
if we want to write this
in exponential form,

00:06:41.600 --> 00:06:46.850
this is equal to e to
the ln of 1 plus 3x.

00:06:46.850 --> 00:06:50.560
This is true of any
positive number.

00:06:50.560 --> 00:06:54.240
Any positive number is e
to the ln of it, because e

00:06:54.240 --> 00:06:57.480
and log are inverse functions.

00:06:57.480 --> 00:07:00.120
So OK, so we write it like this.

00:07:00.120 --> 00:07:06.710
And so that means that 1
plus 3x to the 10 over x

00:07:06.710 --> 00:07:16.550
is equal to e to the ln of
1 plus 3x to the 10 over x.

00:07:16.550 --> 00:07:19.970
And now you can use
your exponent rules.

00:07:19.970 --> 00:07:32.510
So this is equal to e to the ln
of 1 plus 3x times 10 over x.

00:07:32.510 --> 00:07:37.430
So our original expression
is equal to this down here.

00:07:37.430 --> 00:07:39.480
So the limit of our
original expression

00:07:39.480 --> 00:07:42.690
is equal to the
limit of this one.

00:07:42.690 --> 00:07:45.240
The other thing to notice is
that because exponentiation

00:07:45.240 --> 00:07:47.230
is a nice, continuous
function, in order

00:07:47.230 --> 00:07:51.500
to compute this limit, or
the limit of this expression,

00:07:51.500 --> 00:07:55.680
it suffices-- just
with base a constant,

00:07:55.680 --> 00:07:59.740
e-- it suffices to compute
the limit of the exponent.

00:07:59.740 --> 00:08:02.020
All right, so let's do that.

00:08:02.020 --> 00:08:04.070
Let's compute the
limit of the exponent.

00:08:04.070 --> 00:08:07.922
So we have the limit,
so it has to be

00:08:07.922 --> 00:08:10.470
x going to the same
place, which in this case

00:08:10.470 --> 00:08:21.205
is to 0 from the right, of ln
of 1 plus 3x times 10 over x.

00:08:21.205 --> 00:08:22.205
Well, this is a product.

00:08:22.205 --> 00:08:23.740
It came to us as a product.

00:08:23.740 --> 00:08:27.540
But there's an obvious way to
rewrite this as a quotient.

00:08:27.540 --> 00:08:30.150
I should say it's an
indeterminate product.

00:08:30.150 --> 00:08:34.250
As x goes to 0, this goes
to ln of 1, which is 0.

00:08:34.250 --> 00:08:37.390
Whereas this goes to infinity,
positive infinity since we're

00:08:37.390 --> 00:08:38.500
coming from the right.

00:08:38.500 --> 00:08:40.790
So this is a 0 times
positive infinity form.

00:08:40.790 --> 00:08:42.882
So it is an indeterminate
limit or sorry

00:08:42.882 --> 00:08:45.840
an indeterminate product.

00:08:48.077 --> 00:08:49.660
And, right, and
there's an obvious way

00:08:49.660 --> 00:08:53.490
to rewrite this as an
indeterminate quotient,

00:08:53.490 --> 00:08:56.600
which is to rewrite
it as the limit

00:08:56.600 --> 00:09:06.770
as x goes 0 plus of
10 ln of 1 plus 3x

00:09:06.770 --> 00:09:09.440
divided by x, as x goes 0.

00:09:09.440 --> 00:09:15.430
So now, this is a limit where
it's an infinity-- sorry--

00:09:15.430 --> 00:09:18.610
a 0 over 0 form.

00:09:18.610 --> 00:09:19.359
So OK, so good.

00:09:19.359 --> 00:09:20.900
So now we can apply
l'Hopital's rule.

00:09:20.900 --> 00:09:25.740
So by l'Hopital's rule,
this is equal to the limit

00:09:25.740 --> 00:09:29.580
as x goes to 0 on
the right of-- well,

00:09:29.580 --> 00:09:35.220
we apply l'Hopital's rule
on the top, we get 10 over 1

00:09:35.220 --> 00:09:40.460
plus 3x, by the
chain rule, times 3.

00:09:40.460 --> 00:09:43.915
And on the bottom,
we just get one.

00:09:43.915 --> 00:09:48.240
So now, OK, so this
is true provided

00:09:48.240 --> 00:09:51.440
this second limit exists.

00:09:51.440 --> 00:09:54.500
And the second limit is
no longer indeterminate.

00:09:54.500 --> 00:09:55.780
It's easy to see what it is.

00:09:55.780 --> 00:09:57.700
You just plug in
x equals 0, and we

00:09:57.700 --> 00:10:04.390
get that this is equal to 30.

00:10:04.390 --> 00:10:10.850
OK so this limit is equal to
30, but this isn't the limit

00:10:10.850 --> 00:10:12.880
that we started out
wanting to compute.

00:10:12.880 --> 00:10:14.920
The limit we started
out wanting to compute

00:10:14.920 --> 00:10:20.990
is e to the ln of 1
plus 3x times 10 over x.

00:10:20.990 --> 00:10:23.340
So it's e to the this.

00:10:23.340 --> 00:10:33.690
So our original limit as x
goes to 0 from the right of 1

00:10:33.690 --> 00:10:39.190
plus 3x to the 10
over x is equal to e

00:10:39.190 --> 00:10:44.989
to the thirtieth power,
which is pretty huge.

00:10:44.989 --> 00:10:53.450
OK, so that's how we--
right-- and OK, so that's

00:10:53.450 --> 00:10:54.950
the answer to our question.

00:10:54.950 --> 00:10:57.160
So we took our
original limit, it

00:10:57.160 --> 00:11:00.020
was this indeterminate
exponential form.

00:11:00.020 --> 00:11:02.260
So what we do to it
is, when you have

00:11:02.260 --> 00:11:03.720
an indeterminate
exponential form,

00:11:03.720 --> 00:11:07.180
you do this rewriting as an
exponential in the base of e

00:11:07.180 --> 00:11:10.040
trick, and then you pass
the limit into the exponent.

00:11:10.040 --> 00:11:11.800
Because e is now
a nice constant,

00:11:11.800 --> 00:11:17.100
life is simple-- life is
simpler I should say-- you pass

00:11:17.100 --> 00:11:18.530
the limit into
the constant, then

00:11:18.530 --> 00:11:22.060
you have an indeterminate ratio,
indeterminate product, which

00:11:22.060 --> 00:11:23.990
you rewrite as an
indeterminate ratio

00:11:23.990 --> 00:11:26.020
on which you can then
apply l'Hopital's rule.

00:11:26.020 --> 00:11:29.730
Or possibly, you know, you
rewrite it as a product.

00:11:29.730 --> 00:11:34.270
Then it's easy to see
what the value is.

00:11:34.270 --> 00:11:35.900
So, all right, so
that's how we deal

00:11:35.900 --> 00:11:38.860
with limits of indeterminate
exponential forms.

00:11:38.860 --> 00:11:40.601
And I'll end there.