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PROFESSOR: So we're through
with techniques of integration,
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00:00:24 --> 00:00:26
which is really the most
technical thing that
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00:00:26 --> 00:00:28
we're going to be doing.
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And now we're just clearing
up a few loose ends
13
00:00:34 --> 00:00:37
about calculus.
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00:00:37 --> 00:00:40
And the one we're going to talk
about today will allow us
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to deal with infinity.
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00:00:45 --> 00:00:50
And it's what's known
as L'Hopital's Rule.
17
00:00:50 --> 00:00:55
Here's L'Hopital's Rule.
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00:00:55 --> 00:01:01
And that's what we're
going to do today.
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00:01:01 --> 00:01:14
L'Hopital's Rule it's also
known as L'Hospital's Rule.
20
00:01:14 --> 00:01:19
That's the same name, since the
circumflex is what you put
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in French to omit the s.
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So it's the same thing,
and it's still pronounced
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L'Hopital, even if
it's got an s in it.
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Alright, so that's the
first thing you need
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00:01:32 --> 00:01:33
to know about it.
26
00:01:33 --> 00:01:37
And what this method does
is, it's a convenient
27
00:01:37 --> 00:01:55
way to calculate limits
including some new ones.
28
00:01:55 --> 00:02:02
So it'll be convenient
for the old ones.
29
00:02:02 --> 00:02:09
There are going to be some new
ones and, as an example, you
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00:02:09 --> 00:02:14
can calculate x ln x as
x goes to infinity.
31
00:02:14 --> 00:02:16
You could, whoops, that's
not a very interesting
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00:02:16 --> 00:02:20
one, let's try x goes to
0 from the positive side.
33
00:02:20 --> 00:02:26
And you can calculate, for
example, x e^ - x, as
34
00:02:26 --> 00:02:30
x goes to infinity.
35
00:02:30 --> 00:02:37
And, well, maybe I should
include a few others.
36
00:02:37 --> 00:02:46
Maybe something like ln x /
x as x goes to infinity.
37
00:02:46 --> 00:02:50
So these are some examples of
things which, in fact, if you
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00:02:50 --> 00:02:52
plug into your calculator,
you can see what's
39
00:02:52 --> 00:02:53
happening with these.
40
00:02:53 --> 00:02:56
But if you want to understand
them systematically, it's much
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00:02:56 --> 00:03:00
better to have this tool
of L'Hopital's Rule.
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00:03:00 --> 00:03:02
And certainly there isn't
a proof just based on a
43
00:03:02 --> 00:03:05
calculation in a calculator.
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00:03:05 --> 00:03:07
So now here's the idea.
45
00:03:07 --> 00:03:11
I'll illustrate the idea
first with an example.
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00:03:11 --> 00:03:13
And then we'll make
it systematic.
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00:03:13 --> 00:03:15
And then we're going
to generalize it.
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00:03:15 --> 00:03:18
We'll make it much more, so
when it includes these new
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00:03:18 --> 00:03:21
limits, there are some little
pieces of trickiness that
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00:03:21 --> 00:03:23
you have to understand.
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00:03:23 --> 00:03:28
So, let's just take an example
that you could have done in the
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very first unit of this class.
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The limit as x goes to 1
of x ^ 10 - 1 / x ^2 - 1.
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So that's a limit that
we could've handled.
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00:03:45 --> 00:03:48
And the thing that's
interesting, I mean, if you
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00:03:48 --> 00:03:50
like this is in this category
that we mentioned at the
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beginning of the course
of interesting limits.
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What's interesting about it is
that if you do this silly
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thing, which is just plug in x
= 1, at x = 1 you're
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going to get 0 / 0.
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And that's what we call
an indeterminate form.
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It's just unclear what it is.
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From that plugging, in
you just can't get it.
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Now, on the other hand, there's
a trick for doing this.
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And this is the trick that
we did at the beginning
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of the class.
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And the idea is I can divide
in the numerator and
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denominator by x - 1.
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So this limit is unchanged, if
I try to cancel the hidden
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factor x - 1 in the
numerator and denominator.
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Now, we can actually carry out
these ratios of polynomials and
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calculate them by long
division in algebra.
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That's very, very long.
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We want to do this
with calculus.
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And we already have.
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We already know that this
ratio is what's called
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a difference quotient.
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And then in the limit, it
tends to the derivative
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of this function.
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00:05:08 --> 00:05:12
So the idea is that this is
actually equal to, in the
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00:05:12 --> 00:05:15
limit, now let's just
study one piece of it.
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00:05:15 --> 00:05:21
So if I have a function f ( x),
which is x ^ 10 - 1, and the
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value at 1 happens to be equal
to 0, then this expression that
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00:05:27 --> 00:05:33
we have, which is in disguise,
this is in disguise the
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00:05:33 --> 00:05:39
difference quotient, tends
to, as x goes to 1, the
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derivative, which is f' ( 1).
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That's what it is.
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00:05:43 --> 00:05:45
So we know what the numerator
goes to, and similarly
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00:05:45 --> 00:05:47
we'll know what the
denominator goes to.
90
00:05:47 --> 00:05:51
But what is that?
91
00:05:51 --> 00:05:56
Well, f ' (x) = 10x ^ 9.
92
00:05:58 --> 00:06:00
So we know what the answer is.
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00:06:00 --> 00:06:03
In the numerator it's 10x ^ 9.
94
00:06:03 --> 00:06:06
In the denominator, it's
going to be 2x, that's the
95
00:06:06 --> 00:06:08
derivative of x ^2 - 1.
96
00:06:08 --> 00:06:13
And then were going to have
to evaluate that at x = 1.
97
00:06:13 --> 00:06:18
And so it's going to
be 10/2, which is 5.
98
00:06:18 --> 00:06:19
So the answer is 5.
99
00:06:19 --> 00:06:23
And it's pretty easy to get
from our techniques and
100
00:06:23 --> 00:06:26
knowledge of derivatives,
using this rather
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00:06:26 --> 00:06:27
clever algebraic trick.
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This business of
dividing by x - 1.
103
00:06:33 --> 00:06:37
What I want to do now is
just carry this method
104
00:06:37 --> 00:06:39
out systematically.
105
00:06:39 --> 00:06:44
And that's going to give
us the approach to what's
106
00:06:44 --> 00:06:45
known as L'Hopital's Rule.
107
00:06:45 --> 00:06:48
What my main subject for today.
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00:06:48 --> 00:06:50
So here's the idea.
109
00:06:50 --> 00:06:53
Suppose we're considering, in
general, a limit as x goes to
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00:06:53 --> 00:06:58
some number a of
f ( x) / g ( x).
111
00:06:58 --> 00:07:02
And suppose it's the bad
case where we can't decide.
112
00:07:02 --> 00:07:09
So it's in determinate.
f ( a) = g ( a) = 0.
113
00:07:09 --> 00:07:11
So it would be 0 / 0.
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00:07:11 --> 00:07:13
Now we're just going to do
exactly the same thing
115
00:07:13 --> 00:07:15
we did over here.
116
00:07:15 --> 00:07:19
Namely, we're going to divide a
numerator and denominator, and
117
00:07:19 --> 00:07:21
we're going to repeat
that argument.
118
00:07:21 --> 00:07:25
So we have here f ( x) / x - a.
119
00:07:25 --> 00:07:30
And g (x) / x - a also.
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00:07:30 --> 00:07:33
I haven't changed anything yet.
121
00:07:33 --> 00:07:38
And now I'm going to write
it in this suggestive form.
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00:07:38 --> 00:07:41
Namely, I'm going to take
separately the limit in the
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00:07:41 --> 00:07:42
numerator and the denominator.
124
00:07:42 --> 00:07:44
And I'm going to make
one more shift.
125
00:07:44 --> 00:07:47
So I'm going to take the
limit, as x goes to a in the
126
00:07:47 --> 00:07:49
numerator, but I'm going to
write it as f ( x)
127
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- f ( a) / x - a.
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00:07:53 --> 00:07:54
So that's the way I'm going to
write the numerator, and I've
129
00:07:54 --> 00:07:57
got to draw a much
longer line here.
130
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So why am I allowed to do that?
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That's because f (a) = 0.
132
00:08:02 --> 00:08:07
So I didn't change this
numerator of the numerator any
133
00:08:07 --> 00:08:12
by subtracting. f ( a) = 0.
134
00:08:12 --> 00:08:19
And I'll do the same thing
to the denominator.
135
00:08:19 --> 00:08:22
Again, g ( a) = 0,
so this is OK.
136
00:08:22 --> 00:08:25
And lo and behold, I know
what these limits are.
137
00:08:25 --> 00:08:34
This is f ' (a) / g '(a).
138
00:08:34 --> 00:08:34
So that's it.
139
00:08:34 --> 00:08:36
That's the technique and
this evaluates the limit.
140
00:08:36 --> 00:08:38
And it's not so difficult.
141
00:08:38 --> 00:08:40
The formula's pretty
straightforward here.
142
00:08:40 --> 00:08:51
And it works, provided
that g ' (a) is not 0.
143
00:08:51 --> 00:08:52
Yeah, question.
144
00:08:52 --> 00:09:05
STUDENT: [INAUDIBLE]
145
00:09:05 --> 00:09:10
PROFESSOR: The question is, is
there a more intuitive way of
146
00:09:10 --> 00:09:14
understanding this procedure.
147
00:09:14 --> 00:09:21
And I think the short
answer is that there are
148
00:09:21 --> 00:09:22
other, similar, ways.
149
00:09:22 --> 00:09:25
I don't consider them
to be more intuitive.
150
00:09:25 --> 00:09:28
I will be mentioning one of
them, which is the idea of
151
00:09:28 --> 00:09:33
linearization, which goes back
to what we did in Unit 2.
152
00:09:33 --> 00:09:35
I think it's very important
to understand all of these,
153
00:09:35 --> 00:09:36
more or less, at once.
154
00:09:36 --> 00:09:39
But I wouldn't claim that any
of these methods is a more
155
00:09:39 --> 00:09:41
intuitive one than the other.
156
00:09:41 --> 00:09:44
But basically what's happening
is, we're looking at the linear
157
00:09:44 --> 00:09:46
approximation to f, at a.
158
00:09:46 --> 00:09:48
And the linear
approximation to g at a.
159
00:09:48 --> 00:09:52
That's what underlies this.
160
00:09:52 --> 00:09:56
So now I get to formulate for
you L'Hopital's Rule at least
161
00:09:56 --> 00:09:59
in what I would call the easy
version or, if you
162
00:09:59 --> 00:10:00
like, Version 1.
163
00:10:00 --> 00:10:10
So here's L'Hopital's Rule.
164
00:10:10 --> 00:10:15
Version 1.
165
00:10:15 --> 00:10:18
It's not going to be quite the
same as what we just did.
166
00:10:18 --> 00:10:20
It's going to be
much, much better.
167
00:10:20 --> 00:10:22
And more useful.
168
00:10:22 --> 00:10:25
And what is going to take
care of is this problem that
169
00:10:25 --> 00:10:29
the denominator is not 0.
170
00:10:29 --> 00:10:31
So now here's what
we're going to do.
171
00:10:31 --> 00:10:35
We're going to say that it
turns out that the limit a x
172
00:10:35 --> 00:10:43
goes to a of f(x) / g ( x) =
the limit as x goes to a
173
00:10:43 --> 00:10:48
to a f '(x) / g' (x).
174
00:10:48 --> 00:10:51
Now, that looks practically the
same as what we said before.
175
00:10:51 --> 00:10:55
And I have to make sure that
you understand when it works.
176
00:10:55 --> 00:11:01
So it works provided
this is one of these
177
00:11:01 --> 00:11:03
undefined expressions.
178
00:11:03 --> 00:11:06
In other words, f (
a) = g ( a) = 0.
179
00:11:06 --> 00:11:11
So we have a 0 / 0
expression, indeterminant.
180
00:11:11 --> 00:11:15
And, also, we need
one more assumption.
181
00:11:15 --> 00:11:30
And the right-hand side, the
right-hand limit exists.
182
00:11:30 --> 00:11:33
Now, this is practically
the same thing as what
183
00:11:33 --> 00:11:35
I said over here.
184
00:11:35 --> 00:11:40
Namely, I took the ratio
of these functions, x
185
00:11:40 --> 00:11:42
^ 10 - 1 and x ^2 - 1.
186
00:11:42 --> 00:11:45
I took their derivatives,
which is what I did
187
00:11:45 --> 00:11:46
right here, right.
188
00:11:46 --> 00:11:48
I just differentiated them
and I took the ratio.
189
00:11:48 --> 00:11:51
This is way easier than the
quotient rule, and is nothing
190
00:11:51 --> 00:11:53
like the quotient rule.
191
00:11:53 --> 00:11:56
Don't think quotient rule.
192
00:11:56 --> 00:11:58
Don't think quotient rule.
193
00:11:58 --> 00:11:59
So we differentiate
the numerator and
194
00:11:59 --> 00:12:02
denominator separately.
195
00:12:02 --> 00:12:08
And then I take the limit as
x goes to 1 and I get 5.
196
00:12:08 --> 00:12:09
So that's what I'm
claiming over here.
197
00:12:09 --> 00:12:12
I take these functions, I
replace them with this ratio of
198
00:12:12 --> 00:12:15
derivatives, and then I take
the limit instead, over here.
199
00:12:15 --> 00:12:18
And it turned out that the
functions got much simpler
200
00:12:18 --> 00:12:19
when I differentiated them.
201
00:12:19 --> 00:12:21
I started with this messy
object and I got this much
202
00:12:21 --> 00:12:25
easier object that I
could easily evaluate.
203
00:12:25 --> 00:12:29
So that's the big game
that's happening here.
204
00:12:29 --> 00:12:33
It works, if this limit makes
sense and this limit exists.
205
00:12:33 --> 00:12:38
Now, notice I didn't claim
that g, that the denominator,
206
00:12:38 --> 00:12:40
had to be non-0.
207
00:12:40 --> 00:12:42
So that's what's going
to help us a little
208
00:12:42 --> 00:12:43
bit in a few examples.
209
00:12:43 --> 00:12:45
So let me give you a couple
of examples and then
210
00:12:45 --> 00:12:46
we'll go further.
211
00:12:46 --> 00:12:48
Now, this is only Version 1.
212
00:12:48 --> 00:12:51
But first we have to understand
how this one works.
213
00:12:51 --> 00:12:56
So here's another example.
214
00:12:56 --> 00:13:06
Take the limit as x goes
to 0, of sin 5x / sin 2x.
215
00:13:06 --> 00:13:11
This is another kind of example
of a limit that we discussed in
216
00:13:11 --> 00:13:12
the first part of the course.
217
00:13:12 --> 00:13:13
Unfortunately, now
we're reviewing stuff.
218
00:13:13 --> 00:13:15
So this should reinforce
what you did there.
219
00:13:15 --> 00:13:20
This will be an easier way
of thinking about it.
220
00:13:20 --> 00:13:25
So by L'Hopital's Rule,
so here's the step.
221
00:13:25 --> 00:13:27
We're going to take
one of these steps.
222
00:13:27 --> 00:13:31
This is the limit, as x goes to
1, of the derivatives here.
223
00:13:31 --> 00:13:43
So that's 5 cos 5x / 2 cos 2x.
224
00:13:43 --> 00:13:46
The limit was 1 over
there, but now it's 0.
225
00:13:46 --> 00:13:48
a is 0 in this case.
226
00:13:48 --> 00:13:51
This is the number a.
227
00:13:51 --> 00:13:54
Thank you.
228
00:13:54 --> 00:13:58
So the limit as x goes to 0
is the same as the limit
229
00:13:58 --> 00:14:01
of the derivatives.
230
00:14:01 --> 00:14:02
And that's easy to evaluate.
231
00:14:02 --> 00:14:04
Cosine of 0 = 1, right.
232
00:14:04 --> 00:14:10
This is equal to 5 (cos (5* 0).
233
00:14:10 --> 00:14:12
And that's a
multiplication sign.
234
00:14:12 --> 00:14:14
Maybe I should just
write this as 0.
235
00:14:14 --> 00:14:17
Divided by 2 cos 0.
236
00:14:17 --> 00:14:24
But you know that that's 5/2.
237
00:14:24 --> 00:14:27
So this is how L'Hopital's
method works.
238
00:14:27 --> 00:14:33
It's pretty painless.
239
00:14:33 --> 00:14:36
I'm going to give you another
example, which shows that it
240
00:14:36 --> 00:14:45
works a little better than the
method that I started out with.
241
00:14:45 --> 00:14:48
Here's what happens if we
consider the function
242
00:14:48 --> 00:14:55
cos x - 1 / x ^2.
243
00:14:55 --> 00:14:57
That was a little
harder to deal with.
244
00:14:57 --> 00:15:03
And again, this is one of these
0 / 0 things near x = 0.
245
00:15:03 --> 00:15:11
As x tends to 0, this goes to
an indeterminate form here.
246
00:15:11 --> 00:15:14
Now, according to our method,
this is equivalent to, now I'm
247
00:15:14 --> 00:15:16
going to use this little wiggle
because I don't want to
248
00:15:16 --> 00:15:20
write limit, limit, limit,
limit a million times.
249
00:15:20 --> 00:15:22
So I'm going to use a
little wiggle here.
250
00:15:22 --> 00:15:26
So as x goes to 0, this is
going to behave the same
251
00:15:26 --> 00:15:30
way as differentiating
numerator and denominator.
252
00:15:30 --> 00:15:33
So again this is going to be
- sin x in the numerator.
253
00:15:33 --> 00:15:42
In the denominator,
it's going to be 2x.
254
00:15:42 --> 00:15:47
Now, notice that we
still haven't won yet.
255
00:15:47 --> 00:15:51
Because this is still
of 0 / 0 type.
256
00:15:51 --> 00:15:54
When you plug in x =
0 you still get 0.
257
00:15:54 --> 00:15:57
But that doesn't
damage the method.
258
00:15:57 --> 00:16:00
That doesn't make
the method fail.
259
00:16:00 --> 00:16:10
This 0 / 0, we can apply
L'Hopital's Rule a second time.
260
00:16:10 --> 00:16:13
And as x goes to 0 this is
the same thing as, again,
261
00:16:13 --> 00:16:14
differentiating the
numerator and denominator.
262
00:16:14 --> 00:16:19
So here I get - cos x in
the numerator, and I get
263
00:16:19 --> 00:16:22
2 in the denominator.
264
00:16:22 --> 00:16:25
Again this is way easier
than differentiating
265
00:16:25 --> 00:16:26
ratios of functions.
266
00:16:26 --> 00:16:28
We're only differentiating
the numerator and the
267
00:16:28 --> 00:16:33
denominator separately.
268
00:16:33 --> 00:16:35
And now this is the end.
269
00:16:35 --> 00:16:48
As x goes to 0, this is -
cos 0 / 2, which is - 1/2.
270
00:16:48 --> 00:16:53
Now, the justification for this
comes only when you win in
271
00:16:53 --> 00:16:56
the end and get the limit.
272
00:16:56 --> 00:16:58
Because what the theorem says
is that if one of these
273
00:16:58 --> 00:17:01
limits exists, then the
preceding one exists.
274
00:17:01 --> 00:17:03
And once the preceding
one exists, then the
275
00:17:03 --> 00:17:03
one before it exists.
276
00:17:03 --> 00:17:09
So once we know that this one
exists, that works backwards.
277
00:17:09 --> 00:17:11
It applies to the preceding
limit, which then applies
278
00:17:11 --> 00:17:15
to the very first one.
279
00:17:15 --> 00:17:18
And the logical structure here
is a little subtle, which is
280
00:17:18 --> 00:17:21
that if the right side exists,
then the left side
281
00:17:21 --> 00:17:25
will also exist.
282
00:17:25 --> 00:17:26
Yeah, question.
283
00:17:26 --> 00:17:32
STUDENT: [INAUDIBLE]
284
00:17:32 --> 00:17:34
PROFESSOR: Why does the
right-hand limit have to exist,
285
00:17:34 --> 00:17:37
isn't it just the derivative
that has to exist?
286
00:17:37 --> 00:17:38
No.
287
00:17:38 --> 00:17:40
The derivative of the
numerator has to exist.
288
00:17:40 --> 00:17:42
The derivative of the
denominator has to exist.
289
00:17:42 --> 00:17:45
And this limit has to exist.
290
00:17:45 --> 00:17:48
What doesn't have to exist, by
the way, I never said that
291
00:17:48 --> 00:17:50
f prime of a has to exist.
292
00:17:50 --> 00:17:53
In fact, it's much,
much more subtle.
293
00:17:53 --> 00:17:56
I'm not claiming that f ' (a)
exists, because in order to
294
00:17:56 --> 00:18:01
evaluate this limit, f
' (a) need not exist.
295
00:18:01 --> 00:18:04
What has to happen is that
nearby, for x not equal to
296
00:18:04 --> 00:18:06
a, these things exist.
297
00:18:06 --> 00:18:09
And then the limit
has to exist.
298
00:18:09 --> 00:18:12
So there's no requirements
that the limits exist.
299
00:18:12 --> 00:18:14
In fact, that's exactly going
to be the point when we
300
00:18:14 --> 00:18:16
evaluate these limits here.
301
00:18:16 --> 00:18:22
Is we don't have to evaluate
it right at the end.
302
00:18:22 --> 00:18:26
STUDENT: [INAUDIBLE]
303
00:18:26 --> 00:18:30
PROFESSOR: So the question that
you're asking is, why is this
304
00:18:30 --> 00:18:31
the hypothesis of the theorem?
305
00:18:31 --> 00:18:34
In other words, why
does this work?
306
00:18:34 --> 00:18:37
Well, the answer is that this
is a theorem that's true.
307
00:18:37 --> 00:18:40
If you drop this hypothesis,
it's totally false.
308
00:18:40 --> 00:18:42
And if you don't have this
hypothesis, you can't use
309
00:18:42 --> 00:18:44
the theorem and you will
get the wrong answer.
310
00:18:44 --> 00:18:48
I mean, it's hard to express
it any further than that.
311
00:18:48 --> 00:18:52
So look, in many cases
we tell you formulas.
312
00:18:52 --> 00:18:55
And in many cases it's so
obvious when they're true
313
00:18:55 --> 00:18:59
that we don't have to
worry about what we say.
314
00:18:59 --> 00:19:01
And indeed, there's
something implicit here.
315
00:19:01 --> 00:19:04
I'm saying well, you know, if I
wrote this symbol down, it must
316
00:19:04 --> 00:19:06
mean that the thing exists.
317
00:19:06 --> 00:19:08
So that's a subtle point.
318
00:19:08 --> 00:19:11
But what I'm emphasizing is
that you don't need to know in
319
00:19:11 --> 00:19:13
advance that this one exists.
320
00:19:13 --> 00:19:18
You do need to know in advance
that that one exists.
321
00:19:18 --> 00:19:19
Essentially, yeah.
322
00:19:19 --> 00:19:24
So that's the direction
that it goes.
323
00:19:24 --> 00:19:28
You can't get away with not
having this exist and still
324
00:19:28 --> 00:19:37
have the statement be true.
325
00:19:37 --> 00:19:38
Alright, another question.
326
00:19:38 --> 00:19:39
Thank you.
327
00:19:39 --> 00:19:47
STUDENT: [INAUDIBLE]
328
00:19:47 --> 00:19:52
PROFESSOR: So I'm getting
a little ahead of myself,
329
00:19:52 --> 00:19:54
but let me just say.
330
00:19:54 --> 00:19:59
In these situations here,
when x is going to 0 and
331
00:19:59 --> 00:20:00
x is going to infinity.
332
00:20:00 --> 00:20:02
For instance, here when x
goes to 0, the logarithm
333
00:20:02 --> 00:20:06
is undefined at x = 0.
334
00:20:06 --> 00:20:08
Nevertheless, this
theorem applies.
335
00:20:08 --> 00:20:10
And we'll be able to use it.
336
00:20:10 --> 00:20:12
Over here, as x goes to
infinity, neither of these --
337
00:20:12 --> 00:20:15
well, actually, come to think
of it, e^ -x, if you like,
338
00:20:15 --> 00:20:17
it's equal to 0 at infinity.
339
00:20:17 --> 00:20:21
If you want to say
that it has a value.
340
00:20:21 --> 00:20:25
But in fact, these expressions
don't necessarily have values.
341
00:20:25 --> 00:20:27
At the ends.
342
00:20:27 --> 00:20:33
And nevertheless, the
theorem applies.
343
00:20:33 --> 00:20:34
I mean, it can exist.
344
00:20:34 --> 00:20:36
It's perfectly OK
for it to exist.
345
00:20:36 --> 00:20:37
It's no problem.
346
00:20:37 --> 00:20:39
It just doesn't need to exists.
347
00:20:39 --> 00:20:45
It isn't forced to exist.
348
00:20:45 --> 00:20:50
So here's a calculation
which we just did.
349
00:20:50 --> 00:20:51
And we evaluated this.
350
00:20:51 --> 00:20:57
Now, I want to make a
comparison with the
351
00:20:57 --> 00:21:06
method of approximation.
352
00:21:06 --> 00:21:11
In the method of
approximations, this Example 2,
353
00:21:11 --> 00:21:15
which was the example with the
sine function, we would use the
354
00:21:15 --> 00:21:16
following property.
355
00:21:16 --> 00:21:19
We would use sin u
is approximately u.
356
00:21:19 --> 00:21:22
We would use that
linear approximation.
357
00:21:22 --> 00:21:29
And then what we would have
here is that sin 5x / sin 2x
358
00:21:29 --> 00:21:35
is approximately 5x / 2x,
which is of course 5/2.
359
00:21:35 --> 00:21:38
And this is true when u is
approximately 0, and this is
360
00:21:38 --> 00:21:45
true certainly as x goes to 0,
it's going to be a valid limit.
361
00:21:45 --> 00:21:50
So that's very similar
to Example 2.
362
00:21:50 --> 00:21:54
In Example 3, we managed to
look at this expression
363
00:21:54 --> 00:21:59
cos x - 1 / x ^2.
364
00:21:59 --> 00:22:03
And for this one, you have to
remember the approximation near
365
00:22:03 --> 00:22:07
x = 0 to the cosine function.
366
00:22:07 --> 00:22:15
And that's 1 - x ^2 / 2.
367
00:22:15 --> 00:22:18
So that was the approximation,
the quadratic approximation
368
00:22:18 --> 00:22:20
to the cosine function.
369
00:22:20 --> 00:22:22
And now, sure enough,
this simplifies.
370
00:22:22 --> 00:22:32
This becomes - x ^2 / 2
/ x ^2, which is - 1/2.
371
00:22:32 --> 00:22:34
So we get the same answer,
which is a good thing.
372
00:22:34 --> 00:22:36
Because both of these
methods are valid.
373
00:22:36 --> 00:22:39
They're consistent.
374
00:22:39 --> 00:22:42
You can see that neither
of them is particularly
375
00:22:42 --> 00:22:42
a lot longer.
376
00:22:42 --> 00:22:45
You may have trouble
remembering this property.
377
00:22:45 --> 00:22:51
But in fact it's something
that you can easily derive.
378
00:22:51 --> 00:22:54
And, indeed, it's related to
the second derivative of the
379
00:22:54 --> 00:22:56
cosine, as is this
calculation here.
380
00:22:56 --> 00:23:04
They're almost the same amount
of numerical content to them.
381
00:23:04 --> 00:23:12
So now what I'd like to do is
explain to you why L'Hopital's
382
00:23:12 --> 00:23:14
Rule works better
in some cases.
383
00:23:14 --> 00:23:20
And the real value that it
has is in handling these
384
00:23:20 --> 00:23:25
other more exotic limits.
385
00:23:25 --> 00:23:33
So now we're going to do
L'Hopital's Rule over again.
386
00:23:33 --> 00:23:35
And I'll handle
these functions.
387
00:23:35 --> 00:23:40
But I'll have to rewrite them,
but we'll just do that.
388
00:23:40 --> 00:23:42
So here's the property.
389
00:23:42 --> 00:23:48
That the limit as x goes to a
of f ( x) / g (x) is equal to
390
00:23:48 --> 00:23:54
the limit as x goes to a
of f ' ( x) / g '(x).
391
00:23:54 --> 00:23:55
That's the property.
392
00:23:55 --> 00:23:57
And this is what we'll
always be using.
393
00:23:57 --> 00:23:59
Very convenient thing.
394
00:23:59 --> 00:24:04
And remember it was
true provided that
395
00:24:04 --> 00:24:11
f ( a) = g (a) = 0.
396
00:24:11 --> 00:24:23
And that the right-hand
side exists.
397
00:24:23 --> 00:24:25
But I claim that it
works better, and I'll
398
00:24:25 --> 00:24:26
get rid of these.
399
00:24:26 --> 00:24:30
But I'll write them again
to show you that it
400
00:24:30 --> 00:24:30
works for these.
401
00:24:30 --> 00:24:43
So there are other cases.
402
00:24:43 --> 00:24:47
And the other cases that
are allowed are this.
403
00:24:47 --> 00:24:51
First of all, as indicated
by what I just erased, you
404
00:24:51 --> 00:24:53
can allow a to be equal to
plus or minus infinity.
405
00:24:53 --> 00:24:57
It's also OK.
406
00:24:57 --> 00:25:04
So you can take the limit
going to the far ends
407
00:25:04 --> 00:25:05
of the universe.
408
00:25:05 --> 00:25:06
Both left and right.
409
00:25:06 --> 00:25:11
And then the other thing that
you can do is, you can allow
410
00:25:11 --> 00:25:19
f ( a) and g (a) to be
plus or minus infinity.
411
00:25:19 --> 00:25:22
Is OK.
412
00:25:22 --> 00:25:25
So now, the point is that
we can handle not just the
413
00:25:25 --> 00:25:33
0 / 0 case, but also the
infinity / infinity case.
414
00:25:33 --> 00:25:36
That's a very powerful
tool, and quite different
415
00:25:36 --> 00:25:42
from the other cases.
416
00:25:42 --> 00:25:49
And the third thing is that the
right-hand side doesn't really
417
00:25:49 --> 00:25:56
quite have to exist, in
the ordinary sense.
418
00:25:56 --> 00:26:00
Or, it could be plus
or minus infinity.
419
00:26:00 --> 00:26:01
That's also OK.
420
00:26:01 --> 00:26:04
That's still information.
421
00:26:04 --> 00:26:10
So if we can see where it
goes, then we're still good.
422
00:26:10 --> 00:26:13
If it goes to plus infinity, if
it goes to 0, if it goes to a
423
00:26:13 --> 00:26:15
finite number, if it goes to
minus infinity, all
424
00:26:15 --> 00:26:16
of that will be OK.
425
00:26:16 --> 00:26:19
It just if it oscillates
wildly that we'll be lost.
426
00:26:19 --> 00:26:27
And those calculations
we'll never encounter.
427
00:26:27 --> 00:26:29
So this basically handles
everything that you could
428
00:26:29 --> 00:26:32
possibly hope for.
429
00:26:32 --> 00:26:37
And it's a very
convenient process.
430
00:26:37 --> 00:26:40
So let me carry out
a few examples.
431
00:26:40 --> 00:26:43
And, let's see, I guess the
first one that I wanted
432
00:26:43 --> 00:26:47
to do was x ln x.
433
00:26:47 --> 00:26:49
So what example are we up to.
434
00:26:49 --> 00:26:57
Example 3, so Example
4 is coming up.
435
00:26:57 --> 00:26:59
Example 4, this is one of the
ones that I wrote at the
436
00:26:59 --> 00:27:06
beginning of the
lecture, x ln x.
437
00:27:06 --> 00:27:12
This one was on our
homework problem.
438
00:27:12 --> 00:27:17
In the limits of
some calculation.
439
00:27:17 --> 00:27:26
But so this one, you have to
look at it first to think
440
00:27:26 --> 00:27:27
about what it's doing.
441
00:27:27 --> 00:27:29
It's an indeterminate form,
but it sort of looks like
442
00:27:29 --> 00:27:30
it's the wrong type.
443
00:27:30 --> 00:27:33
So why is it in an
indeterminate form.
444
00:27:33 --> 00:27:38
This one goes to 0, and this
one goes to minus infinity.
445
00:27:38 --> 00:27:40
So, excuse me, this
is a product.
446
00:27:40 --> 00:27:46
It's 0 times minus infinity.
447
00:27:46 --> 00:27:48
So that's an indeterminate
form, because we don't know
448
00:27:48 --> 00:27:50
whether the 0 wins or the
infinity this could keep
449
00:27:50 --> 00:27:51
getting smaller and smaller and
smaller, and this could be
450
00:27:51 --> 00:27:52
getting bigger and
bigger bigger.
451
00:27:52 --> 00:27:55
The product could be
anything in between.
452
00:27:55 --> 00:27:57
We just don't know.
453
00:27:57 --> 00:28:01
So the first step is to
write this as a ratio
454
00:28:01 --> 00:28:06
of things, rather than
a product of things.
455
00:28:06 --> 00:28:09
And it turns out that the way
to do that is to use the
456
00:28:09 --> 00:28:14
logarithm in the numerator, and
the 1 / x in the denominator.
457
00:28:14 --> 00:28:18
So this is a choice
that I'm making here.
458
00:28:18 --> 00:28:23
Now, I've just converted it to
a limit of the type minus
459
00:28:23 --> 00:28:28
infinity divided by infinity.
460
00:28:28 --> 00:28:30
Because the numerator is going
to minus infinity as x goes to
461
00:28:30 --> 00:28:37
0 plus and the denominator 1 /
x is going to plus infinity.
462
00:28:37 --> 00:28:40
Again, there's a competitions,
but now it's one of the forms
463
00:28:40 --> 00:28:44
to which L'Hopital's
Rule applies.
464
00:28:44 --> 00:28:49
Now I'm just going to
apply L'Hopital's Rule.
465
00:28:49 --> 00:28:54
And what it says is that
I differentiate here.
466
00:28:54 --> 00:28:56
So I just differentiate a
numerator and denominator.
467
00:28:56 --> 00:28:58
Applying L'Hopital's
Rule is a breeze.
468
00:28:58 --> 00:29:03
You just differentiate,
differentiate.
469
00:29:03 --> 00:29:06
And now it just simplifies
and we're done.
470
00:29:06 --> 00:29:12
This is the limit as
x goes to 0 plus of,
471
00:29:12 --> 00:29:14
well, the x^2's cancel.
472
00:29:14 --> 00:29:20
This is the same as just
- x. x factors cancel.
473
00:29:20 --> 00:29:21
And so that's 0.
474
00:29:21 --> 00:29:24
The answer is that it's 0.
475
00:29:24 --> 00:29:30
So x goes to 0 faster then ln
n goes to minus infinity.
476
00:29:30 --> 00:29:36
This 0 was the winner.
477
00:29:36 --> 00:29:44
Something you can't necessarily
predict in advance.
478
00:29:44 --> 00:29:49
So let's do the other two
examples that I wrote down.
479
00:29:49 --> 00:29:53
I'm going to do them in
slightly more generality,
480
00:29:53 --> 00:29:58
because they're the most
fundamental rate properties
481
00:29:58 --> 00:30:01
that you're going to need to
know for the next section.
482
00:30:01 --> 00:30:03
Which is improper integrals.
483
00:30:03 --> 00:30:06
And also they're just very
important for physical
484
00:30:06 --> 00:30:10
math, and any other kind
of thing, basically.
485
00:30:10 --> 00:30:12
So here, let's just do these.
486
00:30:12 --> 00:30:16
So let's see, which one
do I want to do first.
487
00:30:16 --> 00:30:21
So I wrote down the limit of
x e^2 - x, but I'm going to
488
00:30:21 --> 00:30:22
make it even more general.
489
00:30:22 --> 00:30:26
I'm going to make it any
negative power here, where p
490
00:30:26 --> 00:30:30
is some positive constant.
491
00:30:30 --> 00:30:35
Now again, this is a product
of functions, not a quotient,
492
00:30:35 --> 00:30:37
a ratio, of functions.
493
00:30:37 --> 00:30:41
And so I need to rewrite it.
494
00:30:41 --> 00:30:50
I'm going to write
it as x / e ^ p x.
495
00:30:50 --> 00:30:52
And now I'm going to apply,
well, so it's of this
496
00:30:52 --> 00:30:58
form infinity / infinity.
497
00:30:58 --> 00:31:00
And now that's the same as
the limit as x goes to
498
00:31:00 --> 00:31:07
infinity of 1 / p e^ px.
499
00:31:07 --> 00:31:08
So where does that go?
500
00:31:08 --> 00:31:10
As x goes to infinity.
501
00:31:10 --> 00:31:12
Now we can decide.
502
00:31:12 --> 00:31:14
The 1 stays where it is.
503
00:31:14 --> 00:31:23
And this, as x goes to
infinity, goes to infinity.
504
00:31:23 --> 00:31:27
So the answer is 0.
505
00:31:27 --> 00:31:48
And the conclusion is that x
grows more slowly then e ^ px.
506
00:31:48 --> 00:31:49
As x goes to infinity.
507
00:31:49 --> 00:31:50
Remember, p is positive
here, of course.
508
00:31:50 --> 00:31:53
It's the increasing
exponentials.
509
00:31:53 --> 00:32:03
Not the decreasing ones.
510
00:32:03 --> 00:32:08
Let's do a variant of this.
511
00:32:08 --> 00:32:10
I'll do it the opposite way.
512
00:32:10 --> 00:32:13
So I'm going to call
this example 5 '.
513
00:32:13 --> 00:32:15
It really doesn't give us any
more information, but it
514
00:32:15 --> 00:32:18
gives you just a little
bit more practice.
515
00:32:18 --> 00:32:27
So suppose I look at things
the other way. e^ px
516
00:32:27 --> 00:32:29
divided by, say, x^ 100.
517
00:32:35 --> 00:32:42
Now, this is an infinity /
infinity example, again.
518
00:32:42 --> 00:32:44
And you can work out
what it's doing.
519
00:32:44 --> 00:32:48
But there are two ways
of thinking about this.
520
00:32:48 --> 00:32:49
There's the slow way
and the fast way.
521
00:32:49 --> 00:32:54
The slow way is to
differentiate this 100 times.
522
00:32:54 --> 00:32:55
That is, right?
523
00:32:55 --> 00:32:58
Apply L'Hopital's Rule over and
over and over and over again.
524
00:32:58 --> 00:33:00
All the way.
525
00:33:00 --> 00:33:02
It's clear that you
could do it, but it's
526
00:33:02 --> 00:33:03
kind of a nuisance.
527
00:33:03 --> 00:33:06
So there's a much
cleverer trick here.
528
00:33:06 --> 00:33:12
Which is to change this to (the
limit as x goes to infinity of
529
00:33:12 --> 00:33:19
the e ^ px / 100 / x) ^ 100.
530
00:33:25 --> 00:33:31
So if you do that, then we
just have one L'Hopital's
531
00:33:31 --> 00:33:34
Rule step here.
532
00:33:34 --> 00:33:44
And that one is that this is
the same as (x goes to infinity
533
00:33:44 --> 00:33:53
of, well it's p / 100 e^
p x / 100 / 1) ^ 100.
534
00:33:55 --> 00:34:02
That's our L'Hopital's step.
535
00:34:02 --> 00:34:07
And of course, that's
(infinity / 1 ) ^ 100.
536
00:34:09 --> 00:34:10
Which is infinity.
537
00:34:10 --> 00:34:14
Now, again I did this in a
slightly different way to
538
00:34:14 --> 00:34:16
show you that it works
with infinity as well.
539
00:34:16 --> 00:34:18
So that was this other case.
540
00:34:18 --> 00:34:21
The right-hand side can
exist, or it can be
541
00:34:21 --> 00:34:22
plus or minus infinity.
542
00:34:22 --> 00:34:25
And that applies to this limit.
543
00:34:25 --> 00:34:27
And therefore, to
the original limit.
544
00:34:27 --> 00:34:39
And the conclusion here is that
e ^ px, p > 0, grows faster
545
00:34:39 --> 00:34:46
than any power of x.
546
00:34:46 --> 00:34:49
I picked x ^ 100, but
obviously it didn't matter
547
00:34:49 --> 00:34:52
what power I picked.
548
00:34:52 --> 00:35:02
The exponents beat
all the powers.
549
00:35:02 --> 00:35:04
So we have one more of the
ones that I gave at the
550
00:35:04 --> 00:35:07
beginning to take care of.
551
00:35:07 --> 00:35:11
And that one is the logarithm.
552
00:35:11 --> 00:35:15
And its behavior at infinity.
553
00:35:15 --> 00:35:18
So I'll do a slightly
variant on that one, too.
554
00:35:18 --> 00:35:25
So we have Example 6, which is
ln x, and instead of dividing
555
00:35:25 --> 00:35:27
by x, I'm going to
divide by x^ 1/3.
556
00:35:27 --> 00:35:30
I could divide by any positive
power of x, we'll just
557
00:35:30 --> 00:35:32
do this example here.
558
00:35:32 --> 00:35:38
So now this, as x goes to
infinity, is of the form
559
00:35:38 --> 00:35:43
infinity / infinity.
560
00:35:43 --> 00:35:46
And so it's equivalent
to what happens when I
561
00:35:46 --> 00:35:49
differentiate numerator and
denominator separately.
562
00:35:49 --> 00:36:00
And that's 1 / x, and here
I have 1/3 x ^ - 2/3.
563
00:36:00 --> 00:36:03
1 / x, and then 1/3 x^ - 2/3.
564
00:36:03 --> 00:36:06
Now, when the dust settles here
and you get your exponents
565
00:36:06 --> 00:36:10
right, we have an x^ - 1, and
this is an x ^ + 2/3, and
566
00:36:10 --> 00:36:12
that's a 1/3 becomes a 3.
567
00:36:12 --> 00:36:19
So this is what it is.
568
00:36:19 --> 00:36:26
And that's equal to 3x ^ - 1/3.
569
00:36:26 --> 00:36:27
Which we can decide.
570
00:36:27 --> 00:36:30
It goes to 0.
571
00:36:30 --> 00:36:37
As x goes to infinity.
572
00:36:37 --> 00:36:49
And so the conclusion is that
ln x grows more slowly as x
573
00:36:49 --> 00:37:08
goes to infinity, than x ^ 1/3
or any positive power of x.
574
00:37:08 --> 00:37:15
So any x ^ p, p
positive, will work.
575
00:37:15 --> 00:37:17
So ln is really slow,
going to infinity.
576
00:37:17 --> 00:37:20
It's very, very gradual.
577
00:37:20 --> 00:37:21
Yeah, question.
578
00:37:21 --> 00:37:45
STUDENT: [INAUDIBLE]
579
00:37:45 --> 00:37:48
PROFESSOR: The question
is, how many hypotheses
580
00:37:48 --> 00:37:50
do you need here?
581
00:37:50 --> 00:37:57
So I said that, and I think
what you were asking is, if I
582
00:37:57 --> 00:38:02
have this hypothesis, can I
also have this hypothesis.
583
00:38:02 --> 00:38:04
That's OK.
584
00:38:04 --> 00:38:08
I can have this hypothesis
combined with this one.
585
00:38:08 --> 00:38:11
I need something about
f (a) and g ( a).
586
00:38:11 --> 00:38:14
I can't assume nothing
about f(a) and g(a).
587
00:38:14 --> 00:38:18
So in other words, I have to be
faced with either an infinity /
588
00:38:18 --> 00:38:24
infinity, or a 0 / 0 situation.
589
00:38:24 --> 00:38:26
So let's see.
590
00:38:26 --> 00:38:35
A rule applies in the 0 / 0,
or infinity / infinity case.
591
00:38:35 --> 00:38:40
These are the only two
cases that it applies in.
592
00:38:40 --> 00:38:45
And a can be anything.
593
00:38:45 --> 00:38:48
Including infinity.
594
00:38:48 --> 00:38:51
Plus or minus infinity.
595
00:38:51 --> 00:38:53
The rule applies in
these two cases.
596
00:38:53 --> 00:38:58
So in other words, this is
what f ( a) / g ( a) is.
597
00:38:58 --> 00:39:00
Either one of these.
598
00:39:00 --> 00:39:02
And in fact, it can
be plus or minus.
599
00:39:02 --> 00:39:06
STUDENT: [INAUDIBLE]
600
00:39:06 --> 00:39:10
PROFESSOR: And the right-hand
side has to be something.
601
00:39:10 --> 00:39:21
It has to be either finite
or plus or minus infinity.
602
00:39:21 --> 00:39:23
So you need something.
603
00:39:23 --> 00:39:26
You need a specific value of
a, you need to decide whether
604
00:39:26 --> 00:39:28
it's an indeterminate form.
605
00:39:28 --> 00:39:30
And you need the right-hand
limit to exist.
606
00:39:30 --> 00:39:33
It's not hard to impose this.
607
00:39:33 --> 00:39:36
Because when you look at the
right-hand side, you'll
608
00:39:36 --> 00:39:37
want to be calculating it.
609
00:39:37 --> 00:39:38
So you want to know what it is.
610
00:39:38 --> 00:39:47
So you'll never have problems
confirming this hypothesis.
611
00:39:47 --> 00:39:51
Alright.
612
00:39:51 --> 00:39:54
Let me give you one
more example here.
613
00:39:54 --> 00:39:56
Which is just
slightly trickier.
614
00:39:56 --> 00:40:15
Which involves, so here's
another indeterminate form.
615
00:40:15 --> 00:40:16
That's going to be 0 ^ 0.
616
00:40:16 --> 00:40:20
617
00:40:20 --> 00:40:22
So there are lots of these
things where you just
618
00:40:22 --> 00:40:23
don't know what to do.
619
00:40:23 --> 00:40:27
And they come out in
various different ways.
620
00:40:27 --> 00:40:32
The simplest example of this
is the limit as x goes to
621
00:40:32 --> 00:40:41
0 from above of x ^ x.
622
00:40:41 --> 00:40:45
In order to work out what's
happening with this one,
623
00:40:45 --> 00:40:47
we have to use a trick.
624
00:40:47 --> 00:40:52
And the trick is this
is a moving exponent.
625
00:40:52 --> 00:40:56
And so it's appropriate
to use base e.
626
00:40:56 --> 00:40:59
This is something that we did
way back in the first unit.
627
00:40:59 --> 00:41:06
So, since we have a
moving exponent, we're
628
00:41:06 --> 00:41:11
going to use base e.
629
00:41:11 --> 00:41:13
That's the good base to
use whenever you have
630
00:41:13 --> 00:41:15
a moving exponent.
631
00:41:15 --> 00:41:21
And so rewrite this as
x^ x = e ^ x ln x.
632
00:41:21 --> 00:41:24
And now, in order to figure out
what's happening, we really
633
00:41:24 --> 00:41:32
only have to know what's
going on with the exponent.
634
00:41:32 --> 00:41:34
So remember, actually
we already did this.
635
00:41:34 --> 00:41:36
But I'm going to do it
once more for you.
636
00:41:36 --> 00:41:39
This is ln x / (1 / x).
637
00:41:39 --> 00:41:44
And that's equivalent, as x
goes to 0, to using L'Hopital's
638
00:41:44 --> 00:41:51
Rule to 1 / x, and this is -
1 / x ^2, which is -
639
00:41:51 --> 00:41:54
x, which goes to 0.
640
00:41:54 --> 00:41:58
As x goes to 0.
641
00:41:58 --> 00:42:01
And so what we have here is
that this one is going to be
642
00:42:01 --> 00:42:06
equivalent to, well, it's
going to tend to what
643
00:42:06 --> 00:42:07
we got over here.
644
00:42:07 --> 00:42:10
It's e ^ 0.
645
00:42:10 --> 00:42:13
That exponent is what we want.
646
00:42:13 --> 00:42:18
As x goes to 0.
647
00:42:18 --> 00:42:27
So that's the answer This
limit happens to be 1.
648
00:42:27 --> 00:42:30
That's actually relatively easy
to do, given all of the power
649
00:42:30 --> 00:42:42
that we have at our hands.
650
00:42:42 --> 00:42:49
Now, let me give you
one more example.
651
00:42:49 --> 00:42:52
Suppose you're trying to
understand the limit
652
00:42:52 --> 00:42:59
of sin x / x ^2.
653
00:42:59 --> 00:43:06
If you apply L'Hopital's
Rule, as x goes to 0, you're
654
00:43:06 --> 00:43:11
going to get cos x / 2x.
655
00:43:11 --> 00:43:19
And if you apply L'Hopital's
Rule again, as x goes to 0,
656
00:43:19 --> 00:43:24
you're going to get
the - sin x / 2.
657
00:43:24 --> 00:43:35
And this, as x goes
to 0, goes to 0.
658
00:43:35 --> 00:43:39
On the other hand, if you look
at the linear approximation
659
00:43:39 --> 00:43:49
method, linear approximation
says that sin x is
660
00:43:49 --> 00:43:55
approximately x near 0.
661
00:43:55 --> 00:43:59
So that should be x / x ^2.
662
00:43:59 --> 00:44:04
Which is 1 / x, which
goes to infinity.
663
00:44:04 --> 00:44:08
As x goes to 0, at least from
one side, minus infinity
664
00:44:08 --> 00:44:13
to the other side.
665
00:44:13 --> 00:44:17
So there's something fishy
going on here, right?
666
00:44:17 --> 00:44:19
So this is fishy.
667
00:44:19 --> 00:44:21
Or maybe this is
fishy, I don't know.
668
00:44:21 --> 00:44:26
So, tell me what's wrong here.
669
00:44:26 --> 00:44:26
Yeah.
670
00:44:26 --> 00:44:37
STUDENT: [INAUDIBLE]
671
00:44:37 --> 00:44:38
PROFESSOR: OK.
672
00:44:38 --> 00:44:43
So the claim is that the second
application of L'Hopital's
673
00:44:43 --> 00:44:51
Rule, this one, is wrong.
674
00:44:51 --> 00:44:54
And that's correct.
675
00:44:54 --> 00:44:56
And this is where you
have to watch out,
676
00:44:56 --> 00:44:58
with L'Hopital's Rule.
677
00:44:58 --> 00:44:59
This is exactly where
you have to watch out.
678
00:44:59 --> 00:45:02
You have to apply the test.
679
00:45:02 --> 00:45:03
Here it's an
indeterminate form.
680
00:45:03 --> 00:45:08
It's 0 / 0 before I
applied the rule.
681
00:45:08 --> 00:45:10
But in order to apply the
rule the second time, it
682
00:45:10 --> 00:45:12
still has to be 0 / 0.
683
00:45:12 --> 00:45:14
But this one isn't.
684
00:45:14 --> 00:45:19
This one is 1 / 0.
685
00:45:19 --> 00:45:20
It's no longer an
indeterminate form.
686
00:45:20 --> 00:45:22
It's actually infinite.
687
00:45:22 --> 00:45:25
Either plus or minus, depending
on the sign of the denominator.
688
00:45:25 --> 00:45:27
Which is just what
this answer is.
689
00:45:27 --> 00:45:30
So the linear
approximation is safe.
690
00:45:30 --> 00:45:35
And we just applied
L'Hopital's Rule wrong.
691
00:45:35 --> 00:45:55
So the moral of the story here
is look before you L'Hop.
692
00:45:55 --> 00:45:58
Alright.
693
00:45:58 --> 00:46:09
Now, let me say one more thing.
694
00:46:09 --> 00:46:22
I need to pile it on just
a little bit, sorry.
695
00:46:22 --> 00:46:36
So don't use it as a crutch.
696
00:46:36 --> 00:46:39
We don't want to just get
ourselves so weak, after being
697
00:46:39 --> 00:46:42
in the hospital for all this
time, that we can't
698
00:46:42 --> 00:46:55
use, I'm sorry.
699
00:46:55 --> 00:47:00
So remember that you shouldn't
have lost your senses.
700
00:47:00 --> 00:47:09
If you have something
like this, so we'll
701
00:47:09 --> 00:47:12
do this one here.
702
00:47:12 --> 00:47:15
Suppose you're trying to
understand what this does
703
00:47:15 --> 00:47:18
as x goes to infinity.
704
00:47:18 --> 00:47:25
Now, you could L'Hopital's Rule
five times, or four times.
705
00:47:25 --> 00:47:30
And get the answer here.
706
00:47:30 --> 00:47:33
But really, you should realize
that the main terms are sitting
707
00:47:33 --> 00:47:34
there right in front of you.
708
00:47:34 --> 00:47:36
And that there's some
algebra that you can
709
00:47:36 --> 00:47:38
do to simplify this.
710
00:47:38 --> 00:47:45
Namely, it's the same as
1 + 2 / x + 1 / x^ 5.
711
00:47:45 --> 00:47:48
712
00:47:48 --> 00:47:51
And then in the denominator,
well, let's see.
713
00:47:51 --> 00:47:53
It's x.
714
00:47:53 --> 00:47:57
So this would be dividing by
1 / x^ 5 in both numerator
715
00:47:57 --> 00:47:58
and denominator.
716
00:47:58 --> 00:48:04
And here you have 1 / x +
2 over, sorry I overshot.
717
00:48:04 --> 00:48:06
But that's OK.
718
00:48:06 --> 00:48:09
2 / x^ 5 here.
719
00:48:09 --> 00:48:12
So these are the main
term, if you like.
720
00:48:12 --> 00:48:18
And it's the same as 1 / 1 /
x, which is the same as x,
721
00:48:18 --> 00:48:21
and it goes to infinity.
722
00:48:21 --> 00:48:22
As x goes to infinity.
723
00:48:22 --> 00:48:26
Or, if you like, much more
simply, just x ^ 5 /
724
00:48:26 --> 00:48:29
x^ 4 is the main term.
725
00:48:29 --> 00:48:30
Which is x.
726
00:48:30 --> 00:48:31
Which goes to infinity.
727
00:48:31 --> 00:48:35
So don't forget your basic
algebra when you're doing
728
00:48:35 --> 00:48:37
this kind of stuff.
729
00:48:37 --> 00:48:40
Use these things and don't
use L'Hopital's Rule.
730
00:48:40 --> 00:48:42
OK, see you next time.
731
00:48:42 --> 00:48:42