WEBVTT
00:00:00.000 --> 00:00:02.230
The following content is
provided under a Creative
00:00:02.230 --> 00:00:02.896
Commons license.
00:00:02.896 --> 00:00:06.110
Your support will help
MIT OpenCourseWare
00:00:06.110 --> 00:00:09.576
continue to offer high quality
educational resources for free.
00:00:09.576 --> 00:00:12.540
To make a donation or to
view additional materials
00:00:12.540 --> 00:00:19.130
from hundreds of MIT courses,
visit MIT OpenCourseWare
00:00:19.130 --> 00:00:21.660
at ocw.mit.edu.
00:00:21.660 --> 00:00:24.755
PROFESSOR: So we're through
with techniques of integration,
00:00:24.755 --> 00:00:26.880
which is really the most
technical thing that we're
00:00:26.880 --> 00:00:28.040
going to be doing.
00:00:28.040 --> 00:00:37.080
And now we're just clearing up
a few loose ends about calculus.
00:00:37.080 --> 00:00:40.090
And the one we're going
to talk about today
00:00:40.090 --> 00:00:45.530
will allow us to
deal with infinity.
00:00:45.530 --> 00:00:50.940
And it's what's known
as L'Hôpital's Rule.
00:00:50.940 --> 00:00:55.990
Here's L'Hôpital's Rule.
00:00:55.990 --> 00:01:01.360
And that's what we're
going to do today.
00:01:01.360 --> 00:01:14.990
L'Hôpital's Rule it's also
known as L'Hospital's Rule.
00:01:14.990 --> 00:01:19.430
That's the same name, since
the circumflex is what you
00:01:19.430 --> 00:01:25.249
put in French to omit the s.
00:01:25.249 --> 00:01:27.290
So it's the same thing,
and it's still pronounced
00:01:27.290 --> 00:01:29.840
L'Hôpital, even if
it's got an s in it.
00:01:29.840 --> 00:01:31.900
Alright, so that's
the first thing
00:01:31.900 --> 00:01:33.650
you need to know about it.
00:01:33.650 --> 00:01:37.110
And what this
method does is, it's
00:01:37.110 --> 00:01:55.410
a convenient way to calculate
limits including some new ones.
00:01:55.410 --> 00:02:02.670
So it'll be convenient
for the old ones.
00:02:02.670 --> 00:02:09.090
There are going to be some
new ones and, as an example,
00:02:09.090 --> 00:02:14.642
you can calculate x ln
x as x goes to infinity.
00:02:14.642 --> 00:02:16.850
You could, whoops, that's
not a very interesting one,
00:02:16.850 --> 00:02:20.470
let's try x goes to 0
from the positive side.
00:02:20.470 --> 00:02:25.950
And you can calculate,
for example, x e^(-x),
00:02:25.950 --> 00:02:30.190
as x goes to infinity.
00:02:30.190 --> 00:02:37.340
And, well, maybe I should
include a few others.
00:02:37.340 --> 00:02:46.550
Maybe something like ln x
/ x as x goes to infinity.
00:02:46.550 --> 00:02:50.286
So these are some examples
of things which, in fact,
00:02:50.286 --> 00:02:51.660
if you plug into
your calculator,
00:02:51.660 --> 00:02:53.326
you can see what's
happening with these.
00:02:53.326 --> 00:02:55.960
But if you want to understand
them systematically,
00:02:55.960 --> 00:03:00.140
it's much better to have this
tool of L'Hôpital's Rule.
00:03:00.140 --> 00:03:02.220
And certainly
there isn't a proof
00:03:02.220 --> 00:03:05.310
just based on a calculation
in a calculator.
00:03:05.310 --> 00:03:07.760
So now here's the idea.
00:03:07.760 --> 00:03:11.750
I'll illustrate the idea
first with an example.
00:03:11.750 --> 00:03:13.380
And then we'll
make it systematic.
00:03:13.380 --> 00:03:15.020
And then we're going
to generalize it.
00:03:15.020 --> 00:03:17.960
We'll make it much
more-- So when
00:03:17.960 --> 00:03:19.470
it includes these
new limits, there
00:03:19.470 --> 00:03:21.000
are some little
pieces of trickiness
00:03:21.000 --> 00:03:23.050
that you have to understand.
00:03:23.050 --> 00:03:27.770
So, let's just take
an example that you
00:03:27.770 --> 00:03:31.070
could have done in the very
first unit of this class.
00:03:31.070 --> 00:03:37.260
The limit as x goes to 1
of (x^10 - 1) / (x^2 - 1).
00:03:41.230 --> 00:03:45.370
So that's a limit that
we could've handled.
00:03:45.370 --> 00:03:47.100
And the thing
that's interesting,
00:03:47.100 --> 00:03:49.229
I mean, if you like this
is in this category,
00:03:49.229 --> 00:03:51.270
that we mentioned at the
beginning of the course,
00:03:51.270 --> 00:03:52.480
of interesting limits.
00:03:52.480 --> 00:03:54.860
What's interesting
about it is that if you
00:03:54.860 --> 00:04:00.580
do this silly thing, which is
just plug in x = 1, at x = 1
00:04:00.580 --> 00:04:02.480
you're going to get 0 / 0.
00:04:02.480 --> 00:04:12.720
And that's what we call
an indeterminate form.
00:04:12.720 --> 00:04:15.340
It's just unclear what it is.
00:04:15.340 --> 00:04:18.150
From that plugging, in
you just can't get it.
00:04:18.150 --> 00:04:21.360
Now, on the other hand,
there's a trick for doing this.
00:04:21.360 --> 00:04:23.680
And this is the
trick that we did
00:04:23.680 --> 00:04:25.350
at the beginning of the class.
00:04:25.350 --> 00:04:32.730
And the idea is I can divide in
the numerator and denominator
00:04:32.730 --> 00:04:36.230
by x - 1.
00:04:36.230 --> 00:04:40.330
So this limit is
unchanged, if I try
00:04:40.330 --> 00:04:45.000
to cancel the hidden factor
x - 1 in the numerator
00:04:45.000 --> 00:04:46.400
and denominator.
00:04:46.400 --> 00:04:51.630
Now, we can actually carry out
these ratios of polynomials
00:04:51.630 --> 00:04:54.580
and calculate them by
long division in algebra.
00:04:54.580 --> 00:04:55.660
That's very, very long.
00:04:55.660 --> 00:04:57.110
We want to do this
with calculus.
00:04:57.110 --> 00:04:58.710
And we already have.
00:04:58.710 --> 00:05:01.530
We already know that
this ratio is what's
00:05:01.530 --> 00:05:03.420
called a difference quotient.
00:05:03.420 --> 00:05:06.200
And then in the limit, it
tends to the derivative
00:05:06.200 --> 00:05:08.340
of this function.
00:05:08.340 --> 00:05:10.620
So the idea is that
this is actually
00:05:10.620 --> 00:05:14.040
equal to, in the
limit, now let's
00:05:14.040 --> 00:05:15.400
just study one piece of it.
00:05:15.400 --> 00:05:20.630
So if I have a function
f(x), which is x^10 - 1,
00:05:20.630 --> 00:05:25.200
and the value at 1
happens to be equal to 0,
00:05:25.200 --> 00:05:31.280
then this expression that we
have, which is in disguise,
00:05:31.280 --> 00:05:36.320
this is in disguise the
difference quotient,
00:05:36.320 --> 00:05:42.870
tends to, as x goes to 1, the
derivative, which is f'(1).
00:05:42.870 --> 00:05:43.620
That's what it is.
00:05:43.620 --> 00:05:45.203
So we know what the
numerator goes to,
00:05:45.203 --> 00:05:47.650
and similarly we'll know
what the denominator goes to.
00:05:47.650 --> 00:05:51.580
But what is that?
00:05:51.580 --> 00:05:58.460
Well, f'(x) = 10x^9.
00:05:58.460 --> 00:06:00.610
So we know what the answer is.
00:06:00.610 --> 00:06:03.550
In the numerator it's 10x^9.
00:06:03.550 --> 00:06:05.340
In the denominator,
it's going to be 2x,
00:06:05.340 --> 00:06:08.340
that's the derivative
of x^2 - 1.
00:06:08.340 --> 00:06:13.930
And then were going to have
to evaluate that at x = 1.
00:06:13.930 --> 00:06:18.160
And so it's going to
be 10/2, which is 5.
00:06:18.160 --> 00:06:19.350
So the answer is 5.
00:06:19.350 --> 00:06:23.570
And it's pretty easy to get from
our techniques and knowledge
00:06:23.570 --> 00:06:27.510
of derivatives, using this
rather clever algebraic trick.
00:06:27.510 --> 00:06:33.540
This business of
dividing by x - 1.
00:06:33.540 --> 00:06:38.030
What I want to do now is
just carry this method out
00:06:38.030 --> 00:06:39.420
systematically.
00:06:39.420 --> 00:06:44.290
And that's going to give us
the approach to what's known
00:06:44.290 --> 00:06:48.860
as L'Hôpital's Rule, what--
my main subject for today.
00:06:48.860 --> 00:06:50.040
So here's the idea.
00:06:50.040 --> 00:06:51.830
Suppose we're
considering, in general,
00:06:51.830 --> 00:06:58.860
a limit as x goes to some
number a of f(x) / g(x).
00:06:58.860 --> 00:07:02.330
And suppose it's the bad
case where we can't decide.
00:07:02.330 --> 00:07:03.850
So it's indeterminate.
00:07:03.850 --> 00:07:09.200
f(a) = g(a) = 0.
00:07:09.200 --> 00:07:11.709
So it would be 0 / 0.
00:07:11.709 --> 00:07:13.750
Now we're just going to
do exactly the same thing
00:07:13.750 --> 00:07:15.520
we did over here.
00:07:15.520 --> 00:07:19.170
Namely, we're going to divide
the numerator and denominator,
00:07:19.170 --> 00:07:21.010
and we're going to
repeat that argument.
00:07:21.010 --> 00:07:25.390
So we have here f(x) / (x-a).
00:07:25.390 --> 00:07:30.240
And g(x), divided by x - a also.
00:07:30.240 --> 00:07:33.200
I haven't changed anything yet.
00:07:33.200 --> 00:07:38.120
And now I'm going to write
it in this suggestive form.
00:07:38.120 --> 00:07:40.789
Namely, I'm going to
take separately the limit
00:07:40.789 --> 00:07:42.330
in the numerator
and the denominator.
00:07:42.330 --> 00:07:44.460
And I'm going to
make one more shift.
00:07:44.460 --> 00:07:46.800
So I'm going to take
the limit, as x goes
00:07:46.800 --> 00:07:51.256
to a in the numerator, but I'm
going to write it as ( (f(x) -
00:07:51.256 --> 00:07:52.516
f(a)) / (x - a).
00:07:52.516 --> 00:07:54.640
So that's the way I'm going
to write the numerator,
00:07:54.640 --> 00:07:57.580
and I've got to draw a
much longer line here.
00:07:57.580 --> 00:07:59.720
So why am I allowed to do that?
00:07:59.720 --> 00:08:02.690
That's because f(a) = 0.
00:08:02.690 --> 00:08:04.340
So I didn't change
this numerator
00:08:04.340 --> 00:08:12.210
of the numerator any by
subtracting that. f(a) = 0.
00:08:12.210 --> 00:08:19.260
And I'll do the same
thing to the denominator.
00:08:19.260 --> 00:08:22.650
Again, g(a) = 0, so this is OK.
00:08:22.650 --> 00:08:25.760
And lo and behold, I know
what these limits are.
00:08:25.760 --> 00:08:27.900
This is f'(a) / g'(a).
00:08:34.110 --> 00:08:34.660
So that's it.
00:08:34.660 --> 00:08:36.864
That's the technique and
this evaluates the limit.
00:08:36.864 --> 00:08:38.530
And it's not so
difficult. The formula's
00:08:38.530 --> 00:08:40.310
pretty straightforward here.
00:08:40.310 --> 00:08:51.410
And it works, provided
that g'(a) is not 0.
00:08:51.410 --> 00:08:52.590
Yeah, question.
00:08:52.590 --> 00:09:05.040
STUDENT: [INAUDIBLE]
00:09:05.040 --> 00:09:09.665
PROFESSOR: The
question is, is there
00:09:09.665 --> 00:09:14.210
a more intuitive way of
understanding this procedure.
00:09:14.210 --> 00:09:21.150
And I think the short
answer is that there
00:09:21.150 --> 00:09:22.600
are other, similar, ways.
00:09:22.600 --> 00:09:25.520
I don't consider them
to be more intuitive.
00:09:25.520 --> 00:09:26.940
I will be mentioning
one of them,
00:09:26.940 --> 00:09:29.610
which is the idea of
linearization, which goes
00:09:29.610 --> 00:09:33.019
back to what we did in Unit 2.
00:09:33.019 --> 00:09:35.310
I think it's very important
to understand all of these,
00:09:35.310 --> 00:09:36.810
more or less, at once.
00:09:36.810 --> 00:09:38.993
But I wouldn't claim
that any of these methods
00:09:38.993 --> 00:09:41.824
is a more intuitive
one than the other.
00:09:41.824 --> 00:09:43.240
But basically
what's happening is,
00:09:43.240 --> 00:09:46.340
we're looking at the linear
approximation to f, at a.
00:09:46.340 --> 00:09:48.980
And the linear
approximation to g at a.
00:09:48.980 --> 00:09:52.880
That's what underlies this.
00:09:52.880 --> 00:09:56.630
So now I get to formulate for
you L'Hôpital's Rule at least
00:09:56.630 --> 00:09:59.650
in what I would call the
easy version or, if you like,
00:09:59.650 --> 00:10:00.520
Version 1.
00:10:00.520 --> 00:10:10.290
So here's L'Hôpital's Rule.
00:10:10.290 --> 00:10:15.290
Version 1.
00:10:15.290 --> 00:10:18.480
It's not going to be quite
the same as what we just did.
00:10:18.480 --> 00:10:20.670
It's going to be
much, much better.
00:10:20.670 --> 00:10:22.856
And more useful.
00:10:22.856 --> 00:10:24.230
And what is going
to take care of
00:10:24.230 --> 00:10:29.730
is this problem that the
denominator is not 0.
00:10:29.730 --> 00:10:31.420
So now here's what
we're going to do.
00:10:31.420 --> 00:10:35.370
We're going to say that it turns
out that the limit as x goes
00:10:35.370 --> 00:10:41.860
to a of f(x) / g(x) is
equal to the limit as x goes
00:10:41.860 --> 00:10:45.190
to a of f'(x) / g'(x).
00:10:48.210 --> 00:10:51.380
Now, that looks practically the
same as what we said before.
00:10:51.380 --> 00:10:55.310
And I have to make sure that
you understand when it works.
00:10:55.310 --> 00:11:03.430
So it works provided this is one
of these undefined expressions.
00:11:03.430 --> 00:11:06.700
In other words, = g(a) = 0.
00:11:06.700 --> 00:11:11.320
So we have a 0 / 0
expression, indeterminate.
00:11:11.320 --> 00:11:15.280
And, also, we need
one more assumption.
00:11:15.280 --> 00:11:30.360
And the right-hand side,
the right-hand limit exists.
00:11:30.360 --> 00:11:33.360
Now, this is practically
the same thing
00:11:33.360 --> 00:11:35.310
as what I said over here.
00:11:35.310 --> 00:11:40.840
Namely, I took the ratio of
these functions, x ^ x^10 -
00:11:40.840 --> 00:11:42.540
1 and x^2 - 1.
00:11:42.540 --> 00:11:44.820
I took their
derivatives, which is
00:11:44.820 --> 00:11:46.180
what I did right here, right.
00:11:46.180 --> 00:11:48.480
I just differentiated
them and I took the ratio.
00:11:48.480 --> 00:11:50.750
This is way easier
than the quotient rule,
00:11:50.750 --> 00:11:53.780
and is nothing like
the quotient rule.
00:11:53.780 --> 00:11:56.900
Don't think quotient rule.
00:11:56.900 --> 00:11:58.070
Don't think quotient rule.
00:11:58.070 --> 00:12:00.360
So we differentiate the
numerator and denominator
00:12:00.360 --> 00:12:02.400
separately.
00:12:02.400 --> 00:12:08.030
And then I take the limit
as x goes to 1 and I get 5.
00:12:08.030 --> 00:12:09.940
So that's what I'm
claiming over here.
00:12:09.940 --> 00:12:11.640
I take these functions,
I replace them
00:12:11.640 --> 00:12:13.350
with this ratio of
derivatives, and then
00:12:13.350 --> 00:12:15.650
I take the limit
instead, over here.
00:12:15.650 --> 00:12:17.996
And it turned out that the
functions got much simpler
00:12:17.996 --> 00:12:19.120
when I differentiated them.
00:12:19.120 --> 00:12:21.200
I started with this
messy object and I
00:12:21.200 --> 00:12:25.520
got this much easier object
that I could easily evaluate.
00:12:25.520 --> 00:12:29.830
So that's the big game
that's happening here.
00:12:29.830 --> 00:12:33.495
It works, if this limit makes
sense and this limit exists.
00:12:33.495 --> 00:12:38.360
Now, notice I didn't claim that
g, that the denominator had
00:12:38.360 --> 00:12:40.260
to be nonzero.
00:12:40.260 --> 00:12:41.790
So that's what's
going to help us
00:12:41.790 --> 00:12:43.430
a little bit in a few examples.
00:12:43.430 --> 00:12:45.487
So let me give you
a couple of examples
00:12:45.487 --> 00:12:46.570
and then we'll go further.
00:12:46.570 --> 00:12:48.470
Now, this is only Version 1.
00:12:48.470 --> 00:12:51.340
But first we have to
understand how this one works.
00:12:51.340 --> 00:12:56.300
So here's another example.
00:12:56.300 --> 00:13:02.860
Take the limit as x goes
to 0, of sin(5x) / sin(2x).
00:13:06.400 --> 00:13:09.320
This is another kind
of example of a limit
00:13:09.320 --> 00:13:12.222
that we discussed in the
first part of the course.
00:13:12.222 --> 00:13:13.930
Unfortunately, now
we're reviewing stuff.
00:13:13.930 --> 00:13:15.980
So this should reinforce
what you did there.
00:13:15.980 --> 00:13:20.330
This will be an easier
way of thinking about it.
00:13:20.330 --> 00:13:25.050
So by L'Hôpital's Rule,
so here's the step.
00:13:25.050 --> 00:13:27.870
We're going to take
one of these steps.
00:13:27.870 --> 00:13:31.560
This is the limit, as x goes
to 1, of the derivatives here.
00:13:31.560 --> 00:13:36.710
So that's 5 cos(5x)
/ (2 cos(2x)).
00:13:43.220 --> 00:13:46.080
The limit was 1 over
there, but now it's
00:13:46.080 --> 00:13:48.530
0. a is 0 in this case.
00:13:48.530 --> 00:13:51.500
This is the number a.
00:13:51.500 --> 00:13:54.930
Thank you.
00:13:54.930 --> 00:13:58.230
So the limit as x
goes to 0 is the same
00:13:58.230 --> 00:14:01.400
as the limit of the derivatives.
00:14:01.400 --> 00:14:02.620
And that's easy to evaluate.
00:14:02.620 --> 00:14:04.910
Cosine of 0 is 1, right.
00:14:04.910 --> 00:14:12.341
This is equal to 5 cos(5*0)--
And that's a multiplication
00:14:12.341 --> 00:14:12.840
sign.
00:14:12.840 --> 00:14:14.660
Maybe I should just
write this as 0.
00:14:14.660 --> 00:14:17.240
Divided by 2 cos 0.
00:14:17.240 --> 00:14:24.440
But you know that that's 5/2.
00:14:24.440 --> 00:14:27.340
So this is how
L'Hopital's method works.
00:14:27.340 --> 00:14:33.660
It's pretty painless.
00:14:33.660 --> 00:14:36.090
I'm going to give you
another example, which
00:14:36.090 --> 00:14:38.520
shows that it works a little
better than the method
00:14:38.520 --> 00:14:45.550
that I started out with.
00:14:45.550 --> 00:14:50.010
Here's what happens if we
consider the function (cos x -
00:14:50.010 --> 00:14:51.430
1) / x^2.
00:14:55.470 --> 00:14:57.940
That was a little
harder to deal with.
00:14:57.940 --> 00:15:03.060
And again, this is one of
these 0 / 0 things near x = 0.
00:15:03.060 --> 00:15:11.200
As x tends to 0, this goes to
an indeterminate form here.
00:15:11.200 --> 00:15:13.166
Now, according to
our method, this
00:15:13.166 --> 00:15:15.540
is equivalent to, now I'm
going to use this little wiggle
00:15:15.540 --> 00:15:18.090
because I don't want to
write limit, limit, limit,
00:15:18.090 --> 00:15:20.030
limit a million times.
00:15:20.030 --> 00:15:22.330
So I'm going to use
a little wiggle here.
00:15:22.330 --> 00:15:26.840
So as x goes to 0, this is
going to behave the same way
00:15:26.840 --> 00:15:30.390
as differentiating
numerator and denominator.
00:15:30.390 --> 00:15:33.640
So again this is going to
be -sin x in the numerator.
00:15:33.640 --> 00:15:42.110
In the denominator,
it's going to be 2x.
00:15:42.110 --> 00:15:47.140
Now, notice that we
still haven't won yet.
00:15:47.140 --> 00:15:51.060
Because this is
still of 0 / 0 type.
00:15:51.060 --> 00:15:54.000
When you plug in x
= 0 you still get 0.
00:15:54.000 --> 00:15:57.270
But that doesn't
damage the method.
00:15:57.270 --> 00:16:00.630
That doesn't make
the method fail.
00:16:00.630 --> 00:16:10.410
This 0 / 0, we can apply
L'Hôpital's Rule a second time.
00:16:10.410 --> 00:16:12.380
And as x goes to 0
this is the same thing
00:16:12.380 --> 00:16:14.860
as, again, differentiating
the numerator and denominator.
00:16:14.860 --> 00:16:18.600
So here I get -cos
x in the numerator,
00:16:18.600 --> 00:16:22.480
and I get 2 in the denominator.
00:16:22.480 --> 00:16:25.000
Again this is way easier
than differentiating
00:16:25.000 --> 00:16:26.260
ratios of functions.
00:16:26.260 --> 00:16:29.240
We're only differentiating the
numerator and the denominator
00:16:29.240 --> 00:16:33.060
separately.
00:16:33.060 --> 00:16:35.960
And now this is the end.
00:16:35.960 --> 00:16:48.660
As x goes to 0, this is -
-cos 0 / 2, which is -1/2.
00:16:48.660 --> 00:16:53.000
Now, the justification
for this comes only
00:16:53.000 --> 00:16:56.130
when you win in the
end and get the limit.
00:16:56.130 --> 00:16:58.820
Because what the theorem says
is that if one of these limits
00:16:58.820 --> 00:17:01.255
exists, then the
preceding one exists.
00:17:01.255 --> 00:17:03.630
And once the preceding one
exists, then the one before it
00:17:03.630 --> 00:17:04.130
exists.
00:17:04.130 --> 00:17:09.480
So once we know that this one
exists, that works backwards.
00:17:09.480 --> 00:17:11.420
It applies to the
preceding limit, which then
00:17:11.420 --> 00:17:15.030
applies to the very first one.
00:17:15.030 --> 00:17:17.660
And the logical structure
here is a little subtle,
00:17:17.660 --> 00:17:19.800
which is that if the
right side exists,
00:17:19.800 --> 00:17:25.450
then the left side
will also exist.
00:17:25.450 --> 00:17:26.380
Yeah, question.
00:17:26.380 --> 00:17:32.620
STUDENT: [INAUDIBLE]
00:17:32.620 --> 00:17:34.950
PROFESSOR: Why does the
right-hand limit have to exist,
00:17:34.950 --> 00:17:37.120
isn't it just the derivative
that has to exist?
00:17:37.120 --> 00:17:38.470
No.
00:17:38.470 --> 00:17:40.352
The derivative of the
numerator has to exist.
00:17:40.352 --> 00:17:42.310
The derivative of the
denominator has to exist.
00:17:42.310 --> 00:17:45.340
And this limit has to exist.
00:17:45.340 --> 00:17:47.280
What doesn't have to
exist, by the way,
00:17:47.280 --> 00:17:50.690
I never said that f
prime of a has to exist.
00:17:50.690 --> 00:17:53.690
In fact, it's much,
much more subtle.
00:17:53.690 --> 00:17:55.710
I'm not claiming
that f'(a) exists,
00:17:55.710 --> 00:17:58.060
because in order to
evaluate this limit,
00:17:58.060 --> 00:18:01.080
f'(a) need not exist.
00:18:01.080 --> 00:18:05.460
What has to happen is that
nearby, for x not equal to a,
00:18:05.460 --> 00:18:06.820
these things exist.
00:18:06.820 --> 00:18:09.530
And then the limit has to exist.
00:18:09.530 --> 00:18:12.060
So there's no requirements
that the limits exist.
00:18:12.060 --> 00:18:14.645
In fact, that's exactly
going to be the point when
00:18:14.645 --> 00:18:16.860
we evaluate these limits here.
00:18:16.860 --> 00:18:22.760
Is we don't have to evaluate
it right at the end.
00:18:22.760 --> 00:18:26.540
STUDENT: [INAUDIBLE]
00:18:26.540 --> 00:18:28.710
PROFESSOR: So the question
that you're asking
00:18:28.710 --> 00:18:31.970
is, why is this the
hypothesis of the theorem?
00:18:31.970 --> 00:18:34.700
In other words,
why does this work?
00:18:34.700 --> 00:18:37.370
Well, the answer is that this
is a theorem that's true.
00:18:37.370 --> 00:18:40.050
If you drop this hypothesis,
it's totally false.
00:18:40.050 --> 00:18:41.700
And if you don't
have this hypothesis,
00:18:41.700 --> 00:18:44.820
you can't use the theorem and
you will get the wrong answer.
00:18:44.820 --> 00:18:48.320
I mean, it's hard to express
it any further than that.
00:18:48.320 --> 00:18:52.040
So look, in many cases
we tell you formulas.
00:18:52.040 --> 00:18:54.070
And in many cases
it's so obvious
00:18:54.070 --> 00:18:56.750
when they're true
that we don't have
00:18:56.750 --> 00:18:59.610
to worry about what we say.
00:18:59.610 --> 00:19:01.980
And indeed, there's
something implicit here.
00:19:01.980 --> 00:19:04.400
I'm saying well, you know,
if I wrote this symbol down,
00:19:04.400 --> 00:19:06.370
it must mean that
the thing exists.
00:19:06.370 --> 00:19:08.280
So that's a subtle point.
00:19:08.280 --> 00:19:10.830
But what I'm
emphasizing is that you
00:19:10.830 --> 00:19:13.730
don't need to know in
advance that this one exists.
00:19:13.730 --> 00:19:18.020
You do need to know in
advance that that one exists.
00:19:18.020 --> 00:19:19.850
Essentially, yeah.
00:19:19.850 --> 00:19:24.620
So that's the
direction that it goes.
00:19:24.620 --> 00:19:27.555
You can't get away with
not having this exist
00:19:27.555 --> 00:19:37.517
and still have the
statement be true.
00:19:37.517 --> 00:19:38.600
Alright, another question.
00:19:38.600 --> 00:19:39.740
Thank you.
00:19:39.740 --> 00:19:47.700
STUDENT: [INAUDIBLE]
00:19:47.700 --> 00:19:52.910
PROFESSOR: So I'm getting
a little ahead of myself,
00:19:52.910 --> 00:19:54.610
but let me just say.
00:19:54.610 --> 00:19:59.246
In these situations here,
when x is going to 0 and x
00:19:59.246 --> 00:20:00.120
is going to infinity.
00:20:00.120 --> 00:20:02.020
For instance, here
when x goes to 0,
00:20:02.020 --> 00:20:06.340
the logarithm is
undefined at x = 0.
00:20:06.340 --> 00:20:08.240
Nevertheless, this
theorem applies.
00:20:08.240 --> 00:20:10.170
And we'll be able to use it.
00:20:10.170 --> 00:20:12.520
Over here, as x goes to
infinity, neither of these--
00:20:12.520 --> 00:20:15.540
well, actually, come to think
of it, e^(-x), if you like,
00:20:15.540 --> 00:20:17.890
it's equal to 0 at infinity.
00:20:17.890 --> 00:20:21.220
If you want to say
that it has a value.
00:20:21.220 --> 00:20:24.530
But in fact, these
expressions don't necessarily
00:20:24.530 --> 00:20:27.340
have values, at the ends.
00:20:27.340 --> 00:20:33.160
And nevertheless,
the theorem applies.
00:20:33.160 --> 00:20:34.690
I mean, it can exist.
00:20:34.690 --> 00:20:36.930
It's perfectly OK
for it to exist.
00:20:36.930 --> 00:20:37.910
It's no problem.
00:20:37.910 --> 00:20:39.220
It just doesn't need to exist.
00:20:39.220 --> 00:20:45.080
It isn't forced to exist.
00:20:45.080 --> 00:20:50.090
So here's a calculation
which we just did.
00:20:50.090 --> 00:20:51.460
And we evaluated this.
00:20:51.460 --> 00:20:57.990
Now, I want to make a
comparison with the method
00:20:57.990 --> 00:21:06.770
of approximation.
00:21:06.770 --> 00:21:11.470
In the method of
approximations, this Example 2,
00:21:11.470 --> 00:21:14.480
which was the example
with the sine function,
00:21:14.480 --> 00:21:16.820
we would use the
following property.
00:21:16.820 --> 00:21:19.370
We would use sin u
is approximately u.
00:21:19.370 --> 00:21:22.120
We would use that
linear approximation.
00:21:22.120 --> 00:21:29.980
And then what we would have here
is that sin(5x) / sin(2x) is
00:21:29.980 --> 00:21:35.610
approximately (5x)/(2x),
which is of course 5/2.
00:21:35.610 --> 00:21:38.050
And this is true when
u is approximately 0,
00:21:38.050 --> 00:21:41.150
and this is true
certainly as x goes to 0,
00:21:41.150 --> 00:21:45.880
it's going to be a valid limit.
00:21:45.880 --> 00:21:50.640
So that's very
similar to Example 2.
00:21:50.640 --> 00:21:57.400
In Example 3, we managed to
look at this expression (cos x -
00:21:57.400 --> 00:21:59.640
1) / x^2.
00:21:59.640 --> 00:22:02.270
And for this one,
you have to remember
00:22:02.270 --> 00:22:07.940
the approximation near x =
0 to the cosine function.
00:22:07.940 --> 00:22:15.580
And that's 1 - x^2 / 2.
00:22:15.580 --> 00:22:18.750
So that was the approximation,
the quadratic approximation
00:22:18.750 --> 00:22:20.250
to the cosine function.
00:22:20.250 --> 00:22:22.780
And now, sure enough,
this simplifies.
00:22:22.780 --> 00:22:32.579
This becomes (-x^2 / 2)
/ x^2, which is -1/2.
00:22:32.579 --> 00:22:34.620
So we get the same answer,
which is a good thing.
00:22:34.620 --> 00:22:36.450
Because both of these
methods are valid.
00:22:36.450 --> 00:22:39.520
They're consistent.
00:22:39.520 --> 00:22:42.940
You can see that neither of them
is particularly a lot longer.
00:22:42.940 --> 00:22:45.490
You may have trouble
remembering this property.
00:22:45.490 --> 00:22:51.050
But in fact it's something
that you can easily derive.
00:22:51.050 --> 00:22:54.270
And, indeed, it's related
to the second derivative
00:22:54.270 --> 00:22:56.540
of the cosine, as is
this calculation here.
00:22:56.540 --> 00:23:04.260
They're almost the same amount
of numerical content to them.
00:23:04.260 --> 00:23:11.650
So now what I'd like to
do is explain to you why
00:23:11.650 --> 00:23:14.970
L'Hôpital's Rule works
better in some cases.
00:23:14.970 --> 00:23:19.210
And the real value
that it has is
00:23:19.210 --> 00:23:25.550
in handling these other
more exotic limits.
00:23:25.550 --> 00:23:33.960
So now we're going to do
L'Hôpital's Rule over again.
00:23:33.960 --> 00:23:35.360
And I'll handle these functions.
00:23:35.360 --> 00:23:40.680
But I'll have to rewrite
them, but we'll just do that.
00:23:40.680 --> 00:23:42.100
So here's the property.
00:23:42.100 --> 00:23:48.130
That the limit as x goes to
a of f(x) / g(x) is equal
00:23:48.130 --> 00:23:54.730
to the limit as x goes
to a of f'(x) / g'(x).
00:23:54.730 --> 00:23:55.980
That's the property.
00:23:55.980 --> 00:23:57.990
And this is what
we'll always be using.
00:23:57.990 --> 00:23:59.490
Very convenient thing.
00:23:59.490 --> 00:24:11.530
And remember it was true
provided that f(a) = g(a) = 0.
00:24:11.530 --> 00:24:23.240
And that the
right-hand side exists.
00:24:23.240 --> 00:24:25.257
But I claim that
it works better,
00:24:25.257 --> 00:24:26.340
and I'll get rid of these.
00:24:26.340 --> 00:24:29.330
But I'll write them
again to show you
00:24:29.330 --> 00:24:30.990
that it works for these.
00:24:30.990 --> 00:24:43.680
So there are other cases.
00:24:43.680 --> 00:24:47.250
And the other cases that
are allowed are this.
00:24:47.250 --> 00:24:50.840
First of all, as indicated
by what I just erased,
00:24:50.840 --> 00:24:53.570
you can allow a to be equal
to plus or minus infinity.
00:24:53.570 --> 00:24:57.680
It's also OK.
00:24:57.680 --> 00:25:04.370
So you can take the limit
going to the far ends
00:25:04.370 --> 00:25:05.110
of the universe.
00:25:05.110 --> 00:25:06.630
Both left and right.
00:25:06.630 --> 00:25:09.780
And then the other
thing that you can do
00:25:09.780 --> 00:25:19.690
is, you can allow f(a) and g(a)
to be plus or minus infinity.
00:25:19.690 --> 00:25:22.540
Is OK.
00:25:22.540 --> 00:25:27.450
So now, the point is that we can
handle not just the 0 / 0 case,
00:25:27.450 --> 00:25:33.780
but also the infinity
/ infinity case.
00:25:33.780 --> 00:25:36.710
That's a very powerful
tool, and quite different
00:25:36.710 --> 00:25:42.050
from the other cases.
00:25:42.050 --> 00:25:49.290
And the third thing is
that the right-hand side
00:25:49.290 --> 00:25:56.500
doesn't really quite have to
exist, in the ordinary sense.
00:25:56.500 --> 00:26:00.460
Or, it could be plus
or minus infinity.
00:26:00.460 --> 00:26:01.970
That's also OK.
00:26:01.970 --> 00:26:04.270
That's still information.
00:26:04.270 --> 00:26:10.564
So if we can see where it
goes, then we're still good.
00:26:10.564 --> 00:26:11.980
If it goes to plus
infinity, if it
00:26:11.980 --> 00:26:13.688
goes to 0, if it goes
to a finite number,
00:26:13.688 --> 00:26:16.390
if it goes to minus infinity,
all of that will be OK.
00:26:16.390 --> 00:26:19.400
It just if it oscillates
wildly that we'll be lost.
00:26:19.400 --> 00:26:27.500
And those calculations
we'll never encounter.
00:26:27.500 --> 00:26:29.310
So this basically
handles everything
00:26:29.310 --> 00:26:32.050
that you could
possibly hope for.
00:26:32.050 --> 00:26:37.050
And it's a very
convenient process.
00:26:37.050 --> 00:26:40.510
So let me carry
out a few examples.
00:26:40.510 --> 00:26:44.760
And, let's see, I guess the
first one that I wanted to do
00:26:44.760 --> 00:26:47.640
was x ln x.
00:26:47.640 --> 00:26:49.300
So what example are we up to.
00:26:49.300 --> 00:26:57.150
Example 3, so Example
4 is coming up.
00:26:57.150 --> 00:26:59.030
Example 4, this
is one of the ones
00:26:59.030 --> 00:27:06.430
that I wrote at the beginning
of the lecture, x ln x.
00:27:06.430 --> 00:27:12.930
This one was on our
homework problem.
00:27:12.930 --> 00:27:17.740
In the limits of
some calculation.
00:27:17.740 --> 00:27:25.729
But so this one, you
have to look at it first
00:27:25.729 --> 00:27:27.020
to think about what it's doing.
00:27:27.020 --> 00:27:28.950
It's an indeterminate
form, but it sort of
00:27:28.950 --> 00:27:30.980
looks like it's the wrong type.
00:27:30.980 --> 00:27:33.070
So why is it in an
indeterminate form.
00:27:33.070 --> 00:27:38.600
This one goes to 0, and this
one goes to minus infinity.
00:27:38.600 --> 00:27:40.460
So, excuse me,
this is a product.
00:27:40.460 --> 00:27:45.999
It's 0 times minus infinity.
00:27:45.999 --> 00:27:47.790
So that's an indeterminate
form, because we
00:27:47.790 --> 00:27:49.730
don't know whether the 0
wins or the infinity this
00:27:49.730 --> 00:27:51.620
could keep getting smaller
and smaller and smaller,
00:27:51.620 --> 00:27:52.890
and this could be getting
bigger and bigger bigger.
00:27:52.890 --> 00:27:55.420
The product could be
anything in between.
00:27:55.420 --> 00:27:57.740
We just don't know.
00:27:57.740 --> 00:28:04.270
So the first step is to write
this as a ratio of things,
00:28:04.270 --> 00:28:06.840
rather than a product of things.
00:28:06.840 --> 00:28:08.450
And it turns out
that the way to do
00:28:08.450 --> 00:28:11.590
that is to use the
logarithm in the numerator,
00:28:11.590 --> 00:28:14.360
and the 1 / x in
the denominator.
00:28:14.360 --> 00:28:18.040
So this is a choice
that I'm making here.
00:28:18.040 --> 00:28:23.800
Now, I've just converted it
to a limit of the type minus
00:28:23.800 --> 00:28:28.100
infinity divided by infinity.
00:28:28.100 --> 00:28:30.760
Because the numerator is going
to minus infinity as x goes
00:28:30.760 --> 00:28:37.550
to 0+ and the denominator 1 /
x is going to plus infinity.
00:28:37.550 --> 00:28:40.300
Again, there's a competition,
but now it's one of the forms
00:28:40.300 --> 00:28:44.210
to which L'Hôpital's
Rule applies.
00:28:44.210 --> 00:28:49.400
Now I'm just going to
apply L'Hôpital's Rule.
00:28:49.400 --> 00:28:54.251
And what it says is that
I differentiate here.
00:28:54.251 --> 00:28:56.500
So I just differentiate the
numerator and denominator.
00:28:56.500 --> 00:28:58.900
Applying L'Hôpital's
Rule is a breeze.
00:28:58.900 --> 00:29:03.710
You just differentiate,
differentiate.
00:29:03.710 --> 00:29:06.810
And now it just
simplifies and we're done.
00:29:06.810 --> 00:29:14.180
This is the limit as x goes to
0+ of, well, the x^2's cancel.
00:29:14.180 --> 00:29:20.360
This is the same as just
-x. x factors cancel.
00:29:20.360 --> 00:29:21.820
And so that's 0.
00:29:21.820 --> 00:29:24.170
The answer is that it's 0.
00:29:24.170 --> 00:29:30.650
So x goes to 0 faster then
ln n goes to minus infinity.
00:29:30.650 --> 00:29:36.400
This 0 was the winner.
00:29:36.400 --> 00:29:44.240
Something you can't
necessarily predict in advance.
00:29:44.240 --> 00:29:49.920
So let's do the other two
examples that I wrote down.
00:29:49.920 --> 00:29:53.230
I'm going to do them in
slightly more generality,
00:29:53.230 --> 00:29:57.480
because they're the
most fundamental rate
00:29:57.480 --> 00:29:59.370
properties that
you're going to need
00:29:59.370 --> 00:30:01.220
to know for the next section.
00:30:01.220 --> 00:30:03.000
Which is improper integrals.
00:30:03.000 --> 00:30:07.390
And also they're just very
important for physical math,
00:30:07.390 --> 00:30:10.390
and any other kind
of thing, basically.
00:30:10.390 --> 00:30:12.340
So here, let's just do these.
00:30:12.340 --> 00:30:16.210
So let's see, which one
do I want to do first.
00:30:16.210 --> 00:30:21.179
So I wrote down the
limit of x e^(-x),
00:30:21.179 --> 00:30:22.970
but I'm going to make
it even more general.
00:30:22.970 --> 00:30:25.830
I'm going to make it
any negative power
00:30:25.830 --> 00:30:30.190
here, where p is some
positive constant.
00:30:30.190 --> 00:30:35.680
Now again, this is a product
of functions, not a quotient,
00:30:35.680 --> 00:30:37.350
a ratio, of functions.
00:30:37.350 --> 00:30:41.010
And so I need to rewrite it.
00:30:41.010 --> 00:30:50.670
I'm going to write
it as x / e^(px). p
00:30:50.670 --> 00:30:52.080
And now I'm going
to apply, well,
00:30:52.080 --> 00:30:58.420
so it's of this form
infinity / infinity.
00:30:58.420 --> 00:31:01.860
And now that's the same as the
limit as x goes to infinity
00:31:01.860 --> 00:31:04.080
of 1 / (p e^(px)).
00:31:07.520 --> 00:31:08.670
So where does that go?
00:31:08.670 --> 00:31:10.250
As x goes to infinity.
00:31:10.250 --> 00:31:12.710
Now we can decide.
00:31:12.710 --> 00:31:14.430
The 1 stays where it is.
00:31:14.430 --> 00:31:23.440
And this, as x goes to
infinity, goes to infinity.
00:31:23.440 --> 00:31:27.080
So the answer is 0.
00:31:27.080 --> 00:31:48.184
And the conclusion is that x
grows more slowly then e^(px).
00:31:48.184 --> 00:31:49.100
As x goes to infinity.
00:31:49.100 --> 00:31:50.980
Remember, p is positive
here, of course.
00:31:50.980 --> 00:31:53.990
It's the increasing
exponentials.
00:31:53.990 --> 00:32:03.080
Not the decreasing ones.
00:32:03.080 --> 00:32:08.530
Let's do a variant of this.
00:32:08.530 --> 00:32:10.690
I'll do it the opposite way.
00:32:10.690 --> 00:32:13.580
So I'm going to call
this Example 5'.
00:32:13.580 --> 00:32:15.550
It really doesn't give
us any more information,
00:32:15.550 --> 00:32:18.280
but it gives you just a
little bit more practice.
00:32:18.280 --> 00:32:26.170
So suppose I look at
things the other way.
00:32:26.170 --> 00:32:35.260
e^(px) divided by, say, x^100.
00:32:35.260 --> 00:32:42.010
Now, this is an infinity
/ infinity example, again.
00:32:42.010 --> 00:32:44.820
And you can work
out what it's doing.
00:32:44.820 --> 00:32:47.897
But there are two ways
of thinking about this.
00:32:47.897 --> 00:32:49.480
There's the slow way
and the fast way.
00:32:49.480 --> 00:32:54.390
The slow way is to
differentiate this 100 times.
00:32:54.390 --> 00:32:55.530
That is, right?
00:32:55.530 --> 00:32:58.490
Apply L'Hôpital's Rule over and
over and over and over again.
00:32:58.490 --> 00:33:00.430
All the way.
00:33:00.430 --> 00:33:03.570
It's clear that you could do
it, but it's kind of a nuisance.
00:33:03.570 --> 00:33:06.850
So there's a much
cleverer trick here.
00:33:06.850 --> 00:33:12.650
Which is to change this to the
limit, as x goes to infinity,
00:33:12.650 --> 00:33:19.610
of the e ^ e^(px/100) /
x, to the 100th power.
00:33:25.850 --> 00:33:32.590
So if you do that, then we just
have one L'Hôpital's Rule step
00:33:32.590 --> 00:33:34.540
here.
00:33:34.540 --> 00:33:43.170
And that one is that this
is the same as, ...as x goes
00:33:43.170 --> 00:33:51.110
to infinity of, well it's
p/100 e^(px/100) divided by 1,
00:33:51.110 --> 00:33:53.270
all to the 100th power.
00:33:55.830 --> 00:34:02.480
That's our L'Hôpital step.
00:34:02.480 --> 00:34:09.310
And of course, that's
(infinity / 1)^100.
00:34:09.310 --> 00:34:10.410
Which is infinity.
00:34:10.410 --> 00:34:13.920
Now, again I did this in
a slightly different way
00:34:13.920 --> 00:34:16.800
to show you that it works
with infinity as well.
00:34:16.800 --> 00:34:18.400
So that was this other case.
00:34:18.400 --> 00:34:20.730
The right-hand side
can exist, or it
00:34:20.730 --> 00:34:22.460
can be plus or minus infinity.
00:34:22.460 --> 00:34:25.240
And that applies to this limit.
00:34:25.240 --> 00:34:27.990
And therefore, to
the original limit.
00:34:27.990 --> 00:34:35.180
And the conclusion here
is that e^(px), p > 0,
00:34:35.180 --> 00:34:46.340
grows faster than
any power of x.
00:34:46.340 --> 00:34:50.290
I picked x^100, but obviously
it didn't matter what power I
00:34:50.290 --> 00:34:52.670
picked.
00:34:52.670 --> 00:35:02.050
The exponents beat
all the powers.
00:35:02.050 --> 00:35:05.120
So we have one more of the ones
that I gave at the beginning
00:35:05.120 --> 00:35:07.780
to take care of.
00:35:07.780 --> 00:35:11.330
And that one is the logarithm.
00:35:11.330 --> 00:35:15.720
And its behavior at infinity.
00:35:15.720 --> 00:35:18.700
So I'll do a slight
variant on that one, too.
00:35:18.700 --> 00:35:24.070
So we have Example
6, which is ln x,
00:35:24.070 --> 00:35:27.780
and instead of dividing by x,
I'm going to divide by x^(1/3).
00:35:27.780 --> 00:35:29.990
I could divide by any
positive power of x,
00:35:29.990 --> 00:35:32.170
we'll just do this example here.
00:35:32.170 --> 00:35:37.080
So now this, as x
goes to infinity,
00:35:37.080 --> 00:35:43.230
is of the form
infinity / infinity.
00:35:43.230 --> 00:35:46.470
And so it's equivalent
to what happens
00:35:46.470 --> 00:35:49.250
when I differentiate numerator
and denominator separately.
00:35:49.250 --> 00:36:00.320
And that's 1 / x, and
here I have 1/3 x^(-2/3).
00:36:00.320 --> 00:36:03.470
1 / x, and then 1/3 x^(-2/3).
00:36:03.470 --> 00:36:06.600
Now, when the dust settles
here and you get your exponents
00:36:06.600 --> 00:36:10.560
right, we have an x^(-1),
and this is an x x^(-2/3),
00:36:10.560 --> 00:36:12.500
and that's a 1/3 becomes a 3.
00:36:12.500 --> 00:36:19.260
So this is what it is.
00:36:19.260 --> 00:36:23.000
And that's equal to 3x^(-1/3).
00:36:26.480 --> 00:36:27.830
Which we can decide.
00:36:27.830 --> 00:36:30.150
It goes to 0.
00:36:30.150 --> 00:36:37.240
As x goes to infinity.
00:36:37.240 --> 00:36:50.200
And so the conclusion is that
ln x grows more slowly as x goes
00:36:50.200 --> 00:37:08.690
to infinity, than x x^(1/3)
or any positive power of x.
00:37:08.690 --> 00:37:15.500
So any x^p, p
positive, will work.
00:37:15.500 --> 00:37:17.670
So log is really slow,
going to infinity.
00:37:17.670 --> 00:37:20.550
It's very, very gradual.
00:37:20.550 --> 00:37:21.420
Yeah, question.
00:37:21.420 --> 00:37:45.970
STUDENT: [INAUDIBLE]
00:37:45.970 --> 00:37:48.230
PROFESSOR: The
question is, how many
00:37:48.230 --> 00:37:50.630
hypotheses do you need here?
00:37:50.630 --> 00:37:57.590
So I said that, and I think
what you were asking is,
00:37:57.590 --> 00:38:02.940
if I have this hypothesis, can
I also have this hypothesis.
00:38:02.940 --> 00:38:04.870
That's OK.
00:38:04.870 --> 00:38:08.840
I can have this hypothesis
combined with this one.
00:38:08.840 --> 00:38:11.280
I need something
about f(a) and g(a).
00:38:11.280 --> 00:38:14.910
I can't assume nothing
about f(a) and g(a).
00:38:14.910 --> 00:38:18.190
So in other words, I have to be
faced with either an infinity /
00:38:18.190 --> 00:38:24.180
infinity, or a 0 / 0 situation.
00:38:24.180 --> 00:38:26.650
So let's see.
00:38:26.650 --> 00:38:35.160
The rule applies in the 0 / 0,
or infinity / infinity case.
00:38:35.160 --> 00:38:40.910
These are the only two
cases that it applies in.
00:38:40.910 --> 00:38:45.370
And a can be anything.
00:38:45.370 --> 00:38:48.880
Including infinity.
00:38:48.880 --> 00:38:51.230
Plus or minus infinity.
00:38:51.230 --> 00:38:53.390
The rule applies
in these two cases.
00:38:53.390 --> 00:38:58.310
So in other words, this
is what f(a) / g(a) is.
00:38:58.310 --> 00:39:00.750
Either one of these.
00:39:00.750 --> 00:39:02.540
And in fact, it can
be plus or minus.
00:39:02.540 --> 00:39:06.460
STUDENT: [INAUDIBLE]
00:39:06.460 --> 00:39:10.290
PROFESSOR: And the right-hand
side has to be something.
00:39:10.290 --> 00:39:21.250
It has to be either finite
or plus or minus infinity.
00:39:21.250 --> 00:39:23.690
So you need something.
00:39:23.690 --> 00:39:25.680
You need a specific
value of a, you
00:39:25.680 --> 00:39:28.070
need to decide whether
it's an indeterminate form.
00:39:28.070 --> 00:39:30.320
And you need the
right-hand limit to exist.
00:39:30.320 --> 00:39:33.530
It's not hard to impose this.
00:39:33.530 --> 00:39:35.975
Because when you look
at the right-hand side,
00:39:35.975 --> 00:39:37.350
you'll want to be
calculating it.
00:39:37.350 --> 00:39:38.641
So you want to know what it is.
00:39:38.641 --> 00:39:47.640
So you'll never have problems
confirming this hypothesis.
00:39:47.640 --> 00:39:51.280
Alright.
00:39:51.280 --> 00:39:54.480
Let me give you one
more example here.
00:39:54.480 --> 00:39:56.600
Which is just slightly trickier.
00:39:56.600 --> 00:40:15.130
Which involves, so here's
another indeterminate form.
00:40:15.130 --> 00:40:16.780
That's going to be 0^0.
00:40:20.351 --> 00:40:22.350
So there are lots of these
things where you just
00:40:22.350 --> 00:40:23.690
don't know what to do.
00:40:23.690 --> 00:40:27.730
And they come out in
various different ways.
00:40:27.730 --> 00:40:32.620
The simplest example of this is
the limit as x goes to 0 from
00:40:32.620 --> 00:40:34.280
above of x^x.
00:40:41.430 --> 00:40:45.120
In order to work out what's
happening with this one,
00:40:45.120 --> 00:40:47.610
we have to use a trick.
00:40:47.610 --> 00:40:52.650
And the trick is this
is a moving exponent.
00:40:52.650 --> 00:40:56.220
And so it's appropriate
to use base e.
00:40:56.220 --> 00:40:59.010
This is something that we did
way back in the first unit.
00:40:59.010 --> 00:41:06.690
So, since we have
a moving exponent,
00:41:06.690 --> 00:41:11.830
we're going to use base e.
00:41:11.830 --> 00:41:13.600
That's the good
base to use whenever
00:41:13.600 --> 00:41:15.600
you have a moving exponent.
00:41:15.600 --> 00:41:18.570
And so rewrite this
as x^x = e^(x ln x).
00:41:21.530 --> 00:41:23.700
And now, in order to figure
out what's happening,
00:41:23.700 --> 00:41:25.250
we really only
have to know what's
00:41:25.250 --> 00:41:32.140
going on with the exponent.
00:41:32.140 --> 00:41:34.100
So remember, actually
we already did this.
00:41:34.100 --> 00:41:36.100
But I'm going to do
it once more for you.
00:41:36.100 --> 00:41:39.340
This is ln x / (1/x).
00:41:39.340 --> 00:41:42.090
And that's equivalent,
as x goes to 0,
00:41:42.090 --> 00:41:50.690
to using L'Hôpital's Rule
to 1/x, and this is -1/x^2,
00:41:50.690 --> 00:41:54.350
which is -x, which goes to 0.
00:41:54.350 --> 00:41:58.030
As x goes to 0.
00:41:58.030 --> 00:42:01.370
And so what we have here
is that this one is going
00:42:01.370 --> 00:42:05.140
to be equivalent
to, well, it's going
00:42:05.140 --> 00:42:07.520
to tend to what
we got over here.
00:42:07.520 --> 00:42:10.170
It's e^0.
00:42:10.170 --> 00:42:13.980
That exponent is what we want.
00:42:13.980 --> 00:42:18.250
As x goes to 0.
00:42:18.250 --> 00:42:27.190
So that's the answer This
limit happens to be 1.
00:42:27.190 --> 00:42:28.940
That's actually
relatively easy to do,
00:42:28.940 --> 00:42:42.700
given all of the power
that we have at our hands.
00:42:42.700 --> 00:42:49.310
Now, let me give you
one more example.
00:42:49.310 --> 00:42:54.280
Suppose you're trying to
understand the limit of sin x /
00:42:54.280 --> 00:42:54.780
x^2.
00:42:59.060 --> 00:43:06.090
If you apply L'Hôpital's
Rule, as x goes to 0,
00:43:06.090 --> 00:43:11.660
you're going to
get cos x / (2x).
00:43:11.660 --> 00:43:19.270
And if you apply L'Hôpital's
Rule again, as x goes to 0,
00:43:19.270 --> 00:43:24.920
you're going to get - sin x / 2.
00:43:24.920 --> 00:43:35.840
And this, as x goes
to 0, goes to 0.
00:43:35.840 --> 00:43:39.820
On the other hand, if you look
at the linear approximation
00:43:39.820 --> 00:43:49.470
method, linear approximation
says that sin x
00:43:49.470 --> 00:43:55.090
is approximately x near 0.
00:43:55.090 --> 00:43:59.560
So that should be x / x^2.
00:43:59.560 --> 00:44:04.900
Which is 1 / x, which
goes to infinity.
00:44:04.900 --> 00:44:08.140
As x goes to 0, at
least from one side,
00:44:08.140 --> 00:44:13.100
minus infinity to
the other side.
00:44:13.100 --> 00:44:17.320
So there's something fishy
going on here, right?
00:44:17.320 --> 00:44:19.330
So this is fishy.
00:44:19.330 --> 00:44:21.520
Or maybe this is
fishy, I don't know.
00:44:21.520 --> 00:44:26.251
So, tell me what's wrong here.
00:44:26.251 --> 00:44:26.750
Yeah.
00:44:26.750 --> 00:44:38.230
STUDENT: [INAUDIBLE]
PROFESSOR: OK.
00:44:38.230 --> 00:44:42.615
So the claim is that
the second application
00:44:42.615 --> 00:44:51.950
of L'Hôpital's Rule,
this one, is wrong.
00:44:51.950 --> 00:44:54.620
And that's correct.
00:44:54.620 --> 00:44:56.700
And this is where you
have to watch out,
00:44:56.700 --> 00:44:58.087
with L'Hôpital's Rule.
00:44:58.087 --> 00:44:59.920
This is exactly where
you have to watch out.
00:44:59.920 --> 00:45:02.300
You have to apply the test.
00:45:02.300 --> 00:45:03.820
Here it's an indeterminate form.
00:45:03.820 --> 00:45:08.260
It's 0 / 0 before
I applied the rule.
00:45:08.260 --> 00:45:10.630
But in order to apply
the rule the second time,
00:45:10.630 --> 00:45:12.410
it still has to be 0 / 0.
00:45:12.410 --> 00:45:14.170
But this one isn't.
00:45:14.170 --> 00:45:19.109
This one is 1 / 0.
00:45:19.109 --> 00:45:20.650
It's no longer an
indeterminate form.
00:45:20.650 --> 00:45:22.895
It's actually infinite.
00:45:22.895 --> 00:45:25.520
Either plus or minus, depending
on the sign of the denominator.
00:45:25.520 --> 00:45:27.970
Which is just what
this answer is.
00:45:27.970 --> 00:45:30.550
So the linear
approximation is safe.
00:45:30.550 --> 00:45:35.160
And we just applied
L'Hôpital's Rule wrong.
00:45:35.160 --> 00:45:55.760
So the moral of the story
here is look before you L'Hôp.
00:45:55.760 --> 00:45:58.260
Alright.
00:45:58.260 --> 00:46:09.850
Now, let me say one more thing.
00:46:09.850 --> 00:46:22.900
I need to pile it on
just a little bit, sorry.
00:46:22.900 --> 00:46:36.700
So don't use it as a crutch.
00:46:36.700 --> 00:46:39.110
We don't want to just
get ourselves so weak,
00:46:39.110 --> 00:46:41.190
after being in the
hospital for all this time,
00:46:41.190 --> 00:46:55.560
that we can't use, I'm sorry.
00:46:55.560 --> 00:47:00.700
So remember that you shouldn't
have lost your senses.
00:47:00.700 --> 00:47:12.210
If you have something like
this, so we'll do this one here.
00:47:12.210 --> 00:47:15.030
Suppose you're trying
to understand what this
00:47:15.030 --> 00:47:18.200
does as x goes to infinity.
00:47:18.200 --> 00:47:25.700
Now, you could apply L'Hôpital's
Rule five times, or four times.
00:47:25.700 --> 00:47:30.040
And get the answer here.
00:47:30.040 --> 00:47:33.570
But really, you should realize
that the main terms are sitting
00:47:33.570 --> 00:47:34.870
there right in front of you.
00:47:34.870 --> 00:47:36.453
And that there's
some algebra that you
00:47:36.453 --> 00:47:38.490
can do to simplify this.
00:47:38.490 --> 00:47:44.480
Namely, it's the same
as 1 + 2/x + 1/x^5.
00:47:48.060 --> 00:47:51.510
And then in the denominator,
well, let's see.
00:47:51.510 --> 00:47:53.110
It's x.
00:47:53.110 --> 00:47:57.574
So this would be dividing
by 1/x^5 in both numerator
00:47:57.574 --> 00:47:58.240
and denominator.
00:47:58.240 --> 00:48:04.410
And here you have 1/x plus
2 over, sorry I overshot.
00:48:04.410 --> 00:48:06.830
But that's OK.
00:48:06.830 --> 00:48:09.110
2/x^5 here.
00:48:09.110 --> 00:48:12.650
So these are the main
terms, if you like.
00:48:12.650 --> 00:48:18.510
And it's the same as 1 /
(1/x), which is the same as x,
00:48:18.510 --> 00:48:21.490
and it goes to infinity.
00:48:21.490 --> 00:48:22.930
As x goes to infinity.
00:48:22.930 --> 00:48:25.130
Or, if you like,
much more simply,
00:48:25.130 --> 00:48:29.230
just x^5 / x^4 is the main term.
00:48:29.230 --> 00:48:30.520
Which is x.
00:48:30.520 --> 00:48:31.590
Which goes to infinity.
00:48:31.590 --> 00:48:35.350
So don't forget
your basic algebra
00:48:35.350 --> 00:48:37.210
when you're doing
this kind of stuff.
00:48:37.210 --> 00:48:40.640
Use these things and don't
use L'Hôpital's Rule.
00:48:40.640 --> 00:48:42.355
OK, see you next time.