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PROFESSOR: So we're through
with techniques of integration,
9
00:00:24,755 --> 00:00:26,880
which is really the most
technical thing that we're
10
00:00:26,880 --> 00:00:28,040
going to be doing.
11
00:00:28,040 --> 00:00:37,080
And now we're just clearing up
a few loose ends about calculus.
12
00:00:37,080 --> 00:00:40,090
And the one we're going
to talk about today
13
00:00:40,090 --> 00:00:45,530
will allow us to
deal with infinity.
14
00:00:45,530 --> 00:00:50,940
And it's what's known
as L'Hôpital's Rule.
15
00:00:50,940 --> 00:00:55,990
Here's L'Hôpital's Rule.
16
00:00:55,990 --> 00:01:01,360
And that's what we're
going to do today.
17
00:01:01,360 --> 00:01:14,990
L'Hôpital's Rule it's also
known as L'Hospital's Rule.
18
00:01:14,990 --> 00:01:19,430
That's the same name, since
the circumflex is what you
19
00:01:19,430 --> 00:01:25,249
put in French to omit the s.
20
00:01:25,249 --> 00:01:27,290
So it's the same thing,
and it's still pronounced
21
00:01:27,290 --> 00:01:29,840
L'Hôpital, even if
it's got an s in it.
22
00:01:29,840 --> 00:01:31,900
Alright, so that's
the first thing
23
00:01:31,900 --> 00:01:33,650
you need to know about it.
24
00:01:33,650 --> 00:01:37,110
And what this
method does is, it's
25
00:01:37,110 --> 00:01:55,410
a convenient way to calculate
limits including some new ones.
26
00:01:55,410 --> 00:02:02,670
So it'll be convenient
for the old ones.
27
00:02:02,670 --> 00:02:09,090
There are going to be some
new ones and, as an example,
28
00:02:09,090 --> 00:02:14,642
you can calculate x ln
x as x goes to infinity.
29
00:02:14,642 --> 00:02:16,850
You could, whoops, that's
not a very interesting one,
30
00:02:16,850 --> 00:02:20,470
let's try x goes to 0
from the positive side.
31
00:02:20,470 --> 00:02:25,950
And you can calculate,
for example, x e^(-x),
32
00:02:25,950 --> 00:02:30,190
as x goes to infinity.
33
00:02:30,190 --> 00:02:37,340
And, well, maybe I should
include a few others.
34
00:02:37,340 --> 00:02:46,550
Maybe something like ln x
/ x as x goes to infinity.
35
00:02:46,550 --> 00:02:50,286
So these are some examples
of things which, in fact,
36
00:02:50,286 --> 00:02:51,660
if you plug into
your calculator,
37
00:02:51,660 --> 00:02:53,326
you can see what's
happening with these.
38
00:02:53,326 --> 00:02:55,960
But if you want to understand
them systematically,
39
00:02:55,960 --> 00:03:00,140
it's much better to have this
tool of L'Hôpital's Rule.
40
00:03:00,140 --> 00:03:02,220
And certainly
there isn't a proof
41
00:03:02,220 --> 00:03:05,310
just based on a calculation
in a calculator.
42
00:03:05,310 --> 00:03:07,760
So now here's the idea.
43
00:03:07,760 --> 00:03:11,750
I'll illustrate the idea
first with an example.
44
00:03:11,750 --> 00:03:13,380
And then we'll
make it systematic.
45
00:03:13,380 --> 00:03:15,020
And then we're going
to generalize it.
46
00:03:15,020 --> 00:03:17,960
We'll make it much
more-- So when
47
00:03:17,960 --> 00:03:19,470
it includes these
new limits, there
48
00:03:19,470 --> 00:03:21,000
are some little
pieces of trickiness
49
00:03:21,000 --> 00:03:23,050
that you have to understand.
50
00:03:23,050 --> 00:03:27,770
So, let's just take
an example that you
51
00:03:27,770 --> 00:03:31,070
could have done in the very
first unit of this class.
52
00:03:31,070 --> 00:03:37,260
The limit as x goes to 1
of (x^10 - 1) / (x^2 - 1).
53
00:03:41,230 --> 00:03:45,370
So that's a limit that
we could've handled.
54
00:03:45,370 --> 00:03:47,100
And the thing
that's interesting,
55
00:03:47,100 --> 00:03:49,229
I mean, if you like this
is in this category,
56
00:03:49,229 --> 00:03:51,270
that we mentioned at the
beginning of the course,
57
00:03:51,270 --> 00:03:52,480
of interesting limits.
58
00:03:52,480 --> 00:03:54,860
What's interesting
about it is that if you
59
00:03:54,860 --> 00:04:00,580
do this silly thing, which is
just plug in x = 1, at x = 1
60
00:04:00,580 --> 00:04:02,480
you're going to get 0 / 0.
61
00:04:02,480 --> 00:04:12,720
And that's what we call
an indeterminate form.
62
00:04:12,720 --> 00:04:15,340
It's just unclear what it is.
63
00:04:15,340 --> 00:04:18,150
From that plugging, in
you just can't get it.
64
00:04:18,150 --> 00:04:21,360
Now, on the other hand,
there's a trick for doing this.
65
00:04:21,360 --> 00:04:23,680
And this is the
trick that we did
66
00:04:23,680 --> 00:04:25,350
at the beginning of the class.
67
00:04:25,350 --> 00:04:32,730
And the idea is I can divide in
the numerator and denominator
68
00:04:32,730 --> 00:04:36,230
by x - 1.
69
00:04:36,230 --> 00:04:40,330
So this limit is
unchanged, if I try
70
00:04:40,330 --> 00:04:45,000
to cancel the hidden factor
x - 1 in the numerator
71
00:04:45,000 --> 00:04:46,400
and denominator.
72
00:04:46,400 --> 00:04:51,630
Now, we can actually carry out
these ratios of polynomials
73
00:04:51,630 --> 00:04:54,580
and calculate them by
long division in algebra.
74
00:04:54,580 --> 00:04:55,660
That's very, very long.
75
00:04:55,660 --> 00:04:57,110
We want to do this
with calculus.
76
00:04:57,110 --> 00:04:58,710
And we already have.
77
00:04:58,710 --> 00:05:01,530
We already know that
this ratio is what's
78
00:05:01,530 --> 00:05:03,420
called a difference quotient.
79
00:05:03,420 --> 00:05:06,200
And then in the limit, it
tends to the derivative
80
00:05:06,200 --> 00:05:08,340
of this function.
81
00:05:08,340 --> 00:05:10,620
So the idea is that
this is actually
82
00:05:10,620 --> 00:05:14,040
equal to, in the
limit, now let's
83
00:05:14,040 --> 00:05:15,400
just study one piece of it.
84
00:05:15,400 --> 00:05:20,630
So if I have a function
f(x), which is x^10 - 1,
85
00:05:20,630 --> 00:05:25,200
and the value at 1
happens to be equal to 0,
86
00:05:25,200 --> 00:05:31,280
then this expression that we
have, which is in disguise,
87
00:05:31,280 --> 00:05:36,320
this is in disguise the
difference quotient,
88
00:05:36,320 --> 00:05:42,870
tends to, as x goes to 1, the
derivative, which is f'(1).
89
00:05:42,870 --> 00:05:43,620
That's what it is.
90
00:05:43,620 --> 00:05:45,203
So we know what the
numerator goes to,
91
00:05:45,203 --> 00:05:47,650
and similarly we'll know
what the denominator goes to.
92
00:05:47,650 --> 00:05:51,580
But what is that?
93
00:05:51,580 --> 00:05:58,460
Well, f'(x) = 10x^9.
94
00:05:58,460 --> 00:06:00,610
So we know what the answer is.
95
00:06:00,610 --> 00:06:03,550
In the numerator it's 10x^9.
96
00:06:03,550 --> 00:06:05,340
In the denominator,
it's going to be 2x,
97
00:06:05,340 --> 00:06:08,340
that's the derivative
of x^2 - 1.
98
00:06:08,340 --> 00:06:13,930
And then were going to have
to evaluate that at x = 1.
99
00:06:13,930 --> 00:06:18,160
And so it's going to
be 10/2, which is 5.
100
00:06:18,160 --> 00:06:19,350
So the answer is 5.
101
00:06:19,350 --> 00:06:23,570
And it's pretty easy to get from
our techniques and knowledge
102
00:06:23,570 --> 00:06:27,510
of derivatives, using this
rather clever algebraic trick.
103
00:06:27,510 --> 00:06:33,540
This business of
dividing by x - 1.
104
00:06:33,540 --> 00:06:38,030
What I want to do now is
just carry this method out
105
00:06:38,030 --> 00:06:39,420
systematically.
106
00:06:39,420 --> 00:06:44,290
And that's going to give us
the approach to what's known
107
00:06:44,290 --> 00:06:48,860
as L'Hôpital's Rule, what--
my main subject for today.
108
00:06:48,860 --> 00:06:50,040
So here's the idea.
109
00:06:50,040 --> 00:06:51,830
Suppose we're
considering, in general,
110
00:06:51,830 --> 00:06:58,860
a limit as x goes to some
number a of f(x) / g(x).
111
00:06:58,860 --> 00:07:02,330
And suppose it's the bad
case where we can't decide.
112
00:07:02,330 --> 00:07:03,850
So it's indeterminate.
113
00:07:03,850 --> 00:07:09,200
f(a) = g(a) = 0.
114
00:07:09,200 --> 00:07:11,709
So it would be 0 / 0.
115
00:07:11,709 --> 00:07:13,750
Now we're just going to
do exactly the same thing
116
00:07:13,750 --> 00:07:15,520
we did over here.
117
00:07:15,520 --> 00:07:19,170
Namely, we're going to divide
the numerator and denominator,
118
00:07:19,170 --> 00:07:21,010
and we're going to
repeat that argument.
119
00:07:21,010 --> 00:07:25,390
So we have here f(x) / (x-a).
120
00:07:25,390 --> 00:07:30,240
And g(x), divided by x - a also.
121
00:07:30,240 --> 00:07:33,200
I haven't changed anything yet.
122
00:07:33,200 --> 00:07:38,120
And now I'm going to write
it in this suggestive form.
123
00:07:38,120 --> 00:07:40,789
Namely, I'm going to
take separately the limit
124
00:07:40,789 --> 00:07:42,330
in the numerator
and the denominator.
125
00:07:42,330 --> 00:07:44,460
And I'm going to
make one more shift.
126
00:07:44,460 --> 00:07:46,800
So I'm going to take
the limit, as x goes
127
00:07:46,800 --> 00:07:51,256
to a in the numerator, but I'm
going to write it as ( (f(x) -
128
00:07:51,256 --> 00:07:52,516
f(a)) / (x - a).
129
00:07:52,516 --> 00:07:54,640
So that's the way I'm going
to write the numerator,
130
00:07:54,640 --> 00:07:57,580
and I've got to draw a
much longer line here.
131
00:07:57,580 --> 00:07:59,720
So why am I allowed to do that?
132
00:07:59,720 --> 00:08:02,690
That's because f(a) = 0.
133
00:08:02,690 --> 00:08:04,340
So I didn't change
this numerator
134
00:08:04,340 --> 00:08:12,210
of the numerator any by
subtracting that. f(a) = 0.
135
00:08:12,210 --> 00:08:19,260
And I'll do the same
thing to the denominator.
136
00:08:19,260 --> 00:08:22,650
Again, g(a) = 0, so this is OK.
137
00:08:22,650 --> 00:08:25,760
And lo and behold, I know
what these limits are.
138
00:08:25,760 --> 00:08:27,900
This is f'(a) / g'(a).
139
00:08:34,110 --> 00:08:34,660
So that's it.
140
00:08:34,660 --> 00:08:36,864
That's the technique and
this evaluates the limit.
141
00:08:36,864 --> 00:08:38,530
And it's not so
difficult. The formula's
142
00:08:38,530 --> 00:08:40,310
pretty straightforward here.
143
00:08:40,310 --> 00:08:51,410
And it works, provided
that g'(a) is not 0.
144
00:08:51,410 --> 00:08:52,590
Yeah, question.
145
00:08:52,590 --> 00:09:05,040
STUDENT: [INAUDIBLE]
146
00:09:05,040 --> 00:09:09,665
PROFESSOR: The
question is, is there
147
00:09:09,665 --> 00:09:14,210
a more intuitive way of
understanding this procedure.
148
00:09:14,210 --> 00:09:21,150
And I think the short
answer is that there
149
00:09:21,150 --> 00:09:22,600
are other, similar, ways.
150
00:09:22,600 --> 00:09:25,520
I don't consider them
to be more intuitive.
151
00:09:25,520 --> 00:09:26,940
I will be mentioning
one of them,
152
00:09:26,940 --> 00:09:29,610
which is the idea of
linearization, which goes
153
00:09:29,610 --> 00:09:33,019
back to what we did in Unit 2.
154
00:09:33,019 --> 00:09:35,310
I think it's very important
to understand all of these,
155
00:09:35,310 --> 00:09:36,810
more or less, at once.
156
00:09:36,810 --> 00:09:38,993
But I wouldn't claim
that any of these methods
157
00:09:38,993 --> 00:09:41,824
is a more intuitive
one than the other.
158
00:09:41,824 --> 00:09:43,240
But basically
what's happening is,
159
00:09:43,240 --> 00:09:46,340
we're looking at the linear
approximation to f, at a.
160
00:09:46,340 --> 00:09:48,980
And the linear
approximation to g at a.
161
00:09:48,980 --> 00:09:52,880
That's what underlies this.
162
00:09:52,880 --> 00:09:56,630
So now I get to formulate for
you L'Hôpital's Rule at least
163
00:09:56,630 --> 00:09:59,650
in what I would call the
easy version or, if you like,
164
00:09:59,650 --> 00:10:00,520
Version 1.
165
00:10:00,520 --> 00:10:10,290
So here's L'Hôpital's Rule.
166
00:10:10,290 --> 00:10:15,290
Version 1.
167
00:10:15,290 --> 00:10:18,480
It's not going to be quite
the same as what we just did.
168
00:10:18,480 --> 00:10:20,670
It's going to be
much, much better.
169
00:10:20,670 --> 00:10:22,856
And more useful.
170
00:10:22,856 --> 00:10:24,230
And what is going
to take care of
171
00:10:24,230 --> 00:10:29,730
is this problem that the
denominator is not 0.
172
00:10:29,730 --> 00:10:31,420
So now here's what
we're going to do.
173
00:10:31,420 --> 00:10:35,370
We're going to say that it turns
out that the limit as x goes
174
00:10:35,370 --> 00:10:41,860
to a of f(x) / g(x) is
equal to the limit as x goes
175
00:10:41,860 --> 00:10:45,190
to a of f'(x) / g'(x).
176
00:10:48,210 --> 00:10:51,380
Now, that looks practically the
same as what we said before.
177
00:10:51,380 --> 00:10:55,310
And I have to make sure that
you understand when it works.
178
00:10:55,310 --> 00:11:03,430
So it works provided this is one
of these undefined expressions.
179
00:11:03,430 --> 00:11:06,700
In other words, = g(a) = 0.
180
00:11:06,700 --> 00:11:11,320
So we have a 0 / 0
expression, indeterminate.
181
00:11:11,320 --> 00:11:15,280
And, also, we need
one more assumption.
182
00:11:15,280 --> 00:11:30,360
And the right-hand side,
the right-hand limit exists.
183
00:11:30,360 --> 00:11:33,360
Now, this is practically
the same thing
184
00:11:33,360 --> 00:11:35,310
as what I said over here.
185
00:11:35,310 --> 00:11:40,840
Namely, I took the ratio of
these functions, x ^ x^10 -
186
00:11:40,840 --> 00:11:42,540
1 and x^2 - 1.
187
00:11:42,540 --> 00:11:44,820
I took their
derivatives, which is
188
00:11:44,820 --> 00:11:46,180
what I did right here, right.
189
00:11:46,180 --> 00:11:48,480
I just differentiated
them and I took the ratio.
190
00:11:48,480 --> 00:11:50,750
This is way easier
than the quotient rule,
191
00:11:50,750 --> 00:11:53,780
and is nothing like
the quotient rule.
192
00:11:53,780 --> 00:11:56,900
Don't think quotient rule.
193
00:11:56,900 --> 00:11:58,070
Don't think quotient rule.
194
00:11:58,070 --> 00:12:00,360
So we differentiate the
numerator and denominator
195
00:12:00,360 --> 00:12:02,400
separately.
196
00:12:02,400 --> 00:12:08,030
And then I take the limit
as x goes to 1 and I get 5.
197
00:12:08,030 --> 00:12:09,940
So that's what I'm
claiming over here.
198
00:12:09,940 --> 00:12:11,640
I take these functions,
I replace them
199
00:12:11,640 --> 00:12:13,350
with this ratio of
derivatives, and then
200
00:12:13,350 --> 00:12:15,650
I take the limit
instead, over here.
201
00:12:15,650 --> 00:12:17,996
And it turned out that the
functions got much simpler
202
00:12:17,996 --> 00:12:19,120
when I differentiated them.
203
00:12:19,120 --> 00:12:21,200
I started with this
messy object and I
204
00:12:21,200 --> 00:12:25,520
got this much easier object
that I could easily evaluate.
205
00:12:25,520 --> 00:12:29,830
So that's the big game
that's happening here.
206
00:12:29,830 --> 00:12:33,495
It works, if this limit makes
sense and this limit exists.
207
00:12:33,495 --> 00:12:38,360
Now, notice I didn't claim that
g, that the denominator had
208
00:12:38,360 --> 00:12:40,260
to be nonzero.
209
00:12:40,260 --> 00:12:41,790
So that's what's
going to help us
210
00:12:41,790 --> 00:12:43,430
a little bit in a few examples.
211
00:12:43,430 --> 00:12:45,487
So let me give you
a couple of examples
212
00:12:45,487 --> 00:12:46,570
and then we'll go further.
213
00:12:46,570 --> 00:12:48,470
Now, this is only Version 1.
214
00:12:48,470 --> 00:12:51,340
But first we have to
understand how this one works.
215
00:12:51,340 --> 00:12:56,300
So here's another example.
216
00:12:56,300 --> 00:13:02,860
Take the limit as x goes
to 0, of sin(5x) / sin(2x).
217
00:13:06,400 --> 00:13:09,320
This is another kind
of example of a limit
218
00:13:09,320 --> 00:13:12,222
that we discussed in the
first part of the course.
219
00:13:12,222 --> 00:13:13,930
Unfortunately, now
we're reviewing stuff.
220
00:13:13,930 --> 00:13:15,980
So this should reinforce
what you did there.
221
00:13:15,980 --> 00:13:20,330
This will be an easier
way of thinking about it.
222
00:13:20,330 --> 00:13:25,050
So by L'Hôpital's Rule,
so here's the step.
223
00:13:25,050 --> 00:13:27,870
We're going to take
one of these steps.
224
00:13:27,870 --> 00:13:31,560
This is the limit, as x goes
to 1, of the derivatives here.
225
00:13:31,560 --> 00:13:36,710
So that's 5 cos(5x)
/ (2 cos(2x)).
226
00:13:43,220 --> 00:13:46,080
The limit was 1 over
there, but now it's
227
00:13:46,080 --> 00:13:48,530
0. a is 0 in this case.
228
00:13:48,530 --> 00:13:51,500
This is the number a.
229
00:13:51,500 --> 00:13:54,930
Thank you.
230
00:13:54,930 --> 00:13:58,230
So the limit as x
goes to 0 is the same
231
00:13:58,230 --> 00:14:01,400
as the limit of the derivatives.
232
00:14:01,400 --> 00:14:02,620
And that's easy to evaluate.
233
00:14:02,620 --> 00:14:04,910
Cosine of 0 is 1, right.
234
00:14:04,910 --> 00:14:12,341
This is equal to 5 cos(5*0)--
And that's a multiplication
235
00:14:12,341 --> 00:14:12,840
sign.
236
00:14:12,840 --> 00:14:14,660
Maybe I should just
write this as 0.
237
00:14:14,660 --> 00:14:17,240
Divided by 2 cos 0.
238
00:14:17,240 --> 00:14:24,440
But you know that that's 5/2.
239
00:14:24,440 --> 00:14:27,340
So this is how
L'Hopital's method works.
240
00:14:27,340 --> 00:14:33,660
It's pretty painless.
241
00:14:33,660 --> 00:14:36,090
I'm going to give you
another example, which
242
00:14:36,090 --> 00:14:38,520
shows that it works a little
better than the method
243
00:14:38,520 --> 00:14:45,550
that I started out with.
244
00:14:45,550 --> 00:14:50,010
Here's what happens if we
consider the function (cos x -
245
00:14:50,010 --> 00:14:51,430
1) / x^2.
246
00:14:55,470 --> 00:14:57,940
That was a little
harder to deal with.
247
00:14:57,940 --> 00:15:03,060
And again, this is one of
these 0 / 0 things near x = 0.
248
00:15:03,060 --> 00:15:11,200
As x tends to 0, this goes to
an indeterminate form here.
249
00:15:11,200 --> 00:15:13,166
Now, according to
our method, this
250
00:15:13,166 --> 00:15:15,540
is equivalent to, now I'm
going to use this little wiggle
251
00:15:15,540 --> 00:15:18,090
because I don't want to
write limit, limit, limit,
252
00:15:18,090 --> 00:15:20,030
limit a million times.
253
00:15:20,030 --> 00:15:22,330
So I'm going to use
a little wiggle here.
254
00:15:22,330 --> 00:15:26,840
So as x goes to 0, this is
going to behave the same way
255
00:15:26,840 --> 00:15:30,390
as differentiating
numerator and denominator.
256
00:15:30,390 --> 00:15:33,640
So again this is going to
be -sin x in the numerator.
257
00:15:33,640 --> 00:15:42,110
In the denominator,
it's going to be 2x.
258
00:15:42,110 --> 00:15:47,140
Now, notice that we
still haven't won yet.
259
00:15:47,140 --> 00:15:51,060
Because this is
still of 0 / 0 type.
260
00:15:51,060 --> 00:15:54,000
When you plug in x
= 0 you still get 0.
261
00:15:54,000 --> 00:15:57,270
But that doesn't
damage the method.
262
00:15:57,270 --> 00:16:00,630
That doesn't make
the method fail.
263
00:16:00,630 --> 00:16:10,410
This 0 / 0, we can apply
L'Hôpital's Rule a second time.
264
00:16:10,410 --> 00:16:12,380
And as x goes to 0
this is the same thing
265
00:16:12,380 --> 00:16:14,860
as, again, differentiating
the numerator and denominator.
266
00:16:14,860 --> 00:16:18,600
So here I get -cos
x in the numerator,
267
00:16:18,600 --> 00:16:22,480
and I get 2 in the denominator.
268
00:16:22,480 --> 00:16:25,000
Again this is way easier
than differentiating
269
00:16:25,000 --> 00:16:26,260
ratios of functions.
270
00:16:26,260 --> 00:16:29,240
We're only differentiating the
numerator and the denominator
271
00:16:29,240 --> 00:16:33,060
separately.
272
00:16:33,060 --> 00:16:35,960
And now this is the end.
273
00:16:35,960 --> 00:16:48,660
As x goes to 0, this is -
-cos 0 / 2, which is -1/2.
274
00:16:48,660 --> 00:16:53,000
Now, the justification
for this comes only
275
00:16:53,000 --> 00:16:56,130
when you win in the
end and get the limit.
276
00:16:56,130 --> 00:16:58,820
Because what the theorem says
is that if one of these limits
277
00:16:58,820 --> 00:17:01,255
exists, then the
preceding one exists.
278
00:17:01,255 --> 00:17:03,630
And once the preceding one
exists, then the one before it
279
00:17:03,630 --> 00:17:04,130
exists.
280
00:17:04,130 --> 00:17:09,480
So once we know that this one
exists, that works backwards.
281
00:17:09,480 --> 00:17:11,420
It applies to the
preceding limit, which then
282
00:17:11,420 --> 00:17:15,030
applies to the very first one.
283
00:17:15,030 --> 00:17:17,660
And the logical structure
here is a little subtle,
284
00:17:17,660 --> 00:17:19,800
which is that if the
right side exists,
285
00:17:19,800 --> 00:17:25,450
then the left side
will also exist.
286
00:17:25,450 --> 00:17:26,380
Yeah, question.
287
00:17:26,380 --> 00:17:32,620
STUDENT: [INAUDIBLE]
288
00:17:32,620 --> 00:17:34,950
PROFESSOR: Why does the
right-hand limit have to exist,
289
00:17:34,950 --> 00:17:37,120
isn't it just the derivative
that has to exist?
290
00:17:37,120 --> 00:17:38,470
No.
291
00:17:38,470 --> 00:17:40,352
The derivative of the
numerator has to exist.
292
00:17:40,352 --> 00:17:42,310
The derivative of the
denominator has to exist.
293
00:17:42,310 --> 00:17:45,340
And this limit has to exist.
294
00:17:45,340 --> 00:17:47,280
What doesn't have to
exist, by the way,
295
00:17:47,280 --> 00:17:50,690
I never said that f
prime of a has to exist.
296
00:17:50,690 --> 00:17:53,690
In fact, it's much,
much more subtle.
297
00:17:53,690 --> 00:17:55,710
I'm not claiming
that f'(a) exists,
298
00:17:55,710 --> 00:17:58,060
because in order to
evaluate this limit,
299
00:17:58,060 --> 00:18:01,080
f'(a) need not exist.
300
00:18:01,080 --> 00:18:05,460
What has to happen is that
nearby, for x not equal to a,
301
00:18:05,460 --> 00:18:06,820
these things exist.
302
00:18:06,820 --> 00:18:09,530
And then the limit has to exist.
303
00:18:09,530 --> 00:18:12,060
So there's no requirements
that the limits exist.
304
00:18:12,060 --> 00:18:14,645
In fact, that's exactly
going to be the point when
305
00:18:14,645 --> 00:18:16,860
we evaluate these limits here.
306
00:18:16,860 --> 00:18:22,760
Is we don't have to evaluate
it right at the end.
307
00:18:22,760 --> 00:18:26,540
STUDENT: [INAUDIBLE]
308
00:18:26,540 --> 00:18:28,710
PROFESSOR: So the question
that you're asking
309
00:18:28,710 --> 00:18:31,970
is, why is this the
hypothesis of the theorem?
310
00:18:31,970 --> 00:18:34,700
In other words,
why does this work?
311
00:18:34,700 --> 00:18:37,370
Well, the answer is that this
is a theorem that's true.
312
00:18:37,370 --> 00:18:40,050
If you drop this hypothesis,
it's totally false.
313
00:18:40,050 --> 00:18:41,700
And if you don't
have this hypothesis,
314
00:18:41,700 --> 00:18:44,820
you can't use the theorem and
you will get the wrong answer.
315
00:18:44,820 --> 00:18:48,320
I mean, it's hard to express
it any further than that.
316
00:18:48,320 --> 00:18:52,040
So look, in many cases
we tell you formulas.
317
00:18:52,040 --> 00:18:54,070
And in many cases
it's so obvious
318
00:18:54,070 --> 00:18:56,750
when they're true
that we don't have
319
00:18:56,750 --> 00:18:59,610
to worry about what we say.
320
00:18:59,610 --> 00:19:01,980
And indeed, there's
something implicit here.
321
00:19:01,980 --> 00:19:04,400
I'm saying well, you know,
if I wrote this symbol down,
322
00:19:04,400 --> 00:19:06,370
it must mean that
the thing exists.
323
00:19:06,370 --> 00:19:08,280
So that's a subtle point.
324
00:19:08,280 --> 00:19:10,830
But what I'm
emphasizing is that you
325
00:19:10,830 --> 00:19:13,730
don't need to know in
advance that this one exists.
326
00:19:13,730 --> 00:19:18,020
You do need to know in
advance that that one exists.
327
00:19:18,020 --> 00:19:19,850
Essentially, yeah.
328
00:19:19,850 --> 00:19:24,620
So that's the
direction that it goes.
329
00:19:24,620 --> 00:19:27,555
You can't get away with
not having this exist
330
00:19:27,555 --> 00:19:37,517
and still have the
statement be true.
331
00:19:37,517 --> 00:19:38,600
Alright, another question.
332
00:19:38,600 --> 00:19:39,740
Thank you.
333
00:19:39,740 --> 00:19:47,700
STUDENT: [INAUDIBLE]
334
00:19:47,700 --> 00:19:52,910
PROFESSOR: So I'm getting
a little ahead of myself,
335
00:19:52,910 --> 00:19:54,610
but let me just say.
336
00:19:54,610 --> 00:19:59,246
In these situations here,
when x is going to 0 and x
337
00:19:59,246 --> 00:20:00,120
is going to infinity.
338
00:20:00,120 --> 00:20:02,020
For instance, here
when x goes to 0,
339
00:20:02,020 --> 00:20:06,340
the logarithm is
undefined at x = 0.
340
00:20:06,340 --> 00:20:08,240
Nevertheless, this
theorem applies.
341
00:20:08,240 --> 00:20:10,170
And we'll be able to use it.
342
00:20:10,170 --> 00:20:12,520
Over here, as x goes to
infinity, neither of these--
343
00:20:12,520 --> 00:20:15,540
well, actually, come to think
of it, e^(-x), if you like,
344
00:20:15,540 --> 00:20:17,890
it's equal to 0 at infinity.
345
00:20:17,890 --> 00:20:21,220
If you want to say
that it has a value.
346
00:20:21,220 --> 00:20:24,530
But in fact, these
expressions don't necessarily
347
00:20:24,530 --> 00:20:27,340
have values, at the ends.
348
00:20:27,340 --> 00:20:33,160
And nevertheless,
the theorem applies.
349
00:20:33,160 --> 00:20:34,690
I mean, it can exist.
350
00:20:34,690 --> 00:20:36,930
It's perfectly OK
for it to exist.
351
00:20:36,930 --> 00:20:37,910
It's no problem.
352
00:20:37,910 --> 00:20:39,220
It just doesn't need to exist.
353
00:20:39,220 --> 00:20:45,080
It isn't forced to exist.
354
00:20:45,080 --> 00:20:50,090
So here's a calculation
which we just did.
355
00:20:50,090 --> 00:20:51,460
And we evaluated this.
356
00:20:51,460 --> 00:20:57,990
Now, I want to make a
comparison with the method
357
00:20:57,990 --> 00:21:06,770
of approximation.
358
00:21:06,770 --> 00:21:11,470
In the method of
approximations, this Example 2,
359
00:21:11,470 --> 00:21:14,480
which was the example
with the sine function,
360
00:21:14,480 --> 00:21:16,820
we would use the
following property.
361
00:21:16,820 --> 00:21:19,370
We would use sin u
is approximately u.
362
00:21:19,370 --> 00:21:22,120
We would use that
linear approximation.
363
00:21:22,120 --> 00:21:29,980
And then what we would have here
is that sin(5x) / sin(2x) is
364
00:21:29,980 --> 00:21:35,610
approximately (5x)/(2x),
which is of course 5/2.
365
00:21:35,610 --> 00:21:38,050
And this is true when
u is approximately 0,
366
00:21:38,050 --> 00:21:41,150
and this is true
certainly as x goes to 0,
367
00:21:41,150 --> 00:21:45,880
it's going to be a valid limit.
368
00:21:45,880 --> 00:21:50,640
So that's very
similar to Example 2.
369
00:21:50,640 --> 00:21:57,400
In Example 3, we managed to
look at this expression (cos x -
370
00:21:57,400 --> 00:21:59,640
1) / x^2.
371
00:21:59,640 --> 00:22:02,270
And for this one,
you have to remember
372
00:22:02,270 --> 00:22:07,940
the approximation near x =
0 to the cosine function.
373
00:22:07,940 --> 00:22:15,580
And that's 1 - x^2 / 2.
374
00:22:15,580 --> 00:22:18,750
So that was the approximation,
the quadratic approximation
375
00:22:18,750 --> 00:22:20,250
to the cosine function.
376
00:22:20,250 --> 00:22:22,780
And now, sure enough,
this simplifies.
377
00:22:22,780 --> 00:22:32,579
This becomes (-x^2 / 2)
/ x^2, which is -1/2.
378
00:22:32,579 --> 00:22:34,620
So we get the same answer,
which is a good thing.
379
00:22:34,620 --> 00:22:36,450
Because both of these
methods are valid.
380
00:22:36,450 --> 00:22:39,520
They're consistent.
381
00:22:39,520 --> 00:22:42,940
You can see that neither of them
is particularly a lot longer.
382
00:22:42,940 --> 00:22:45,490
You may have trouble
remembering this property.
383
00:22:45,490 --> 00:22:51,050
But in fact it's something
that you can easily derive.
384
00:22:51,050 --> 00:22:54,270
And, indeed, it's related
to the second derivative
385
00:22:54,270 --> 00:22:56,540
of the cosine, as is
this calculation here.
386
00:22:56,540 --> 00:23:04,260
They're almost the same amount
of numerical content to them.
387
00:23:04,260 --> 00:23:11,650
So now what I'd like to
do is explain to you why
388
00:23:11,650 --> 00:23:14,970
L'Hôpital's Rule works
better in some cases.
389
00:23:14,970 --> 00:23:19,210
And the real value
that it has is
390
00:23:19,210 --> 00:23:25,550
in handling these other
more exotic limits.
391
00:23:25,550 --> 00:23:33,960
So now we're going to do
L'Hôpital's Rule over again.
392
00:23:33,960 --> 00:23:35,360
And I'll handle these functions.
393
00:23:35,360 --> 00:23:40,680
But I'll have to rewrite
them, but we'll just do that.
394
00:23:40,680 --> 00:23:42,100
So here's the property.
395
00:23:42,100 --> 00:23:48,130
That the limit as x goes to
a of f(x) / g(x) is equal
396
00:23:48,130 --> 00:23:54,730
to the limit as x goes
to a of f'(x) / g'(x).
397
00:23:54,730 --> 00:23:55,980
That's the property.
398
00:23:55,980 --> 00:23:57,990
And this is what
we'll always be using.
399
00:23:57,990 --> 00:23:59,490
Very convenient thing.
400
00:23:59,490 --> 00:24:11,530
And remember it was true
provided that f(a) = g(a) = 0.
401
00:24:11,530 --> 00:24:23,240
And that the
right-hand side exists.
402
00:24:23,240 --> 00:24:25,257
But I claim that
it works better,
403
00:24:25,257 --> 00:24:26,340
and I'll get rid of these.
404
00:24:26,340 --> 00:24:29,330
But I'll write them
again to show you
405
00:24:29,330 --> 00:24:30,990
that it works for these.
406
00:24:30,990 --> 00:24:43,680
So there are other cases.
407
00:24:43,680 --> 00:24:47,250
And the other cases that
are allowed are this.
408
00:24:47,250 --> 00:24:50,840
First of all, as indicated
by what I just erased,
409
00:24:50,840 --> 00:24:53,570
you can allow a to be equal
to plus or minus infinity.
410
00:24:53,570 --> 00:24:57,680
It's also OK.
411
00:24:57,680 --> 00:25:04,370
So you can take the limit
going to the far ends
412
00:25:04,370 --> 00:25:05,110
of the universe.
413
00:25:05,110 --> 00:25:06,630
Both left and right.
414
00:25:06,630 --> 00:25:09,780
And then the other
thing that you can do
415
00:25:09,780 --> 00:25:19,690
is, you can allow f(a) and g(a)
to be plus or minus infinity.
416
00:25:19,690 --> 00:25:22,540
Is OK.
417
00:25:22,540 --> 00:25:27,450
So now, the point is that we can
handle not just the 0 / 0 case,
418
00:25:27,450 --> 00:25:33,780
but also the infinity
/ infinity case.
419
00:25:33,780 --> 00:25:36,710
That's a very powerful
tool, and quite different
420
00:25:36,710 --> 00:25:42,050
from the other cases.
421
00:25:42,050 --> 00:25:49,290
And the third thing is
that the right-hand side
422
00:25:49,290 --> 00:25:56,500
doesn't really quite have to
exist, in the ordinary sense.
423
00:25:56,500 --> 00:26:00,460
Or, it could be plus
or minus infinity.
424
00:26:00,460 --> 00:26:01,970
That's also OK.
425
00:26:01,970 --> 00:26:04,270
That's still information.
426
00:26:04,270 --> 00:26:10,564
So if we can see where it
goes, then we're still good.
427
00:26:10,564 --> 00:26:11,980
If it goes to plus
infinity, if it
428
00:26:11,980 --> 00:26:13,688
goes to 0, if it goes
to a finite number,
429
00:26:13,688 --> 00:26:16,390
if it goes to minus infinity,
all of that will be OK.
430
00:26:16,390 --> 00:26:19,400
It just if it oscillates
wildly that we'll be lost.
431
00:26:19,400 --> 00:26:27,500
And those calculations
we'll never encounter.
432
00:26:27,500 --> 00:26:29,310
So this basically
handles everything
433
00:26:29,310 --> 00:26:32,050
that you could
possibly hope for.
434
00:26:32,050 --> 00:26:37,050
And it's a very
convenient process.
435
00:26:37,050 --> 00:26:40,510
So let me carry
out a few examples.
436
00:26:40,510 --> 00:26:44,760
And, let's see, I guess the
first one that I wanted to do
437
00:26:44,760 --> 00:26:47,640
was x ln x.
438
00:26:47,640 --> 00:26:49,300
So what example are we up to.
439
00:26:49,300 --> 00:26:57,150
Example 3, so Example
4 is coming up.
440
00:26:57,150 --> 00:26:59,030
Example 4, this
is one of the ones
441
00:26:59,030 --> 00:27:06,430
that I wrote at the beginning
of the lecture, x ln x.
442
00:27:06,430 --> 00:27:12,930
This one was on our
homework problem.
443
00:27:12,930 --> 00:27:17,740
In the limits of
some calculation.
444
00:27:17,740 --> 00:27:25,729
But so this one, you
have to look at it first
445
00:27:25,729 --> 00:27:27,020
to think about what it's doing.
446
00:27:27,020 --> 00:27:28,950
It's an indeterminate
form, but it sort of
447
00:27:28,950 --> 00:27:30,980
looks like it's the wrong type.
448
00:27:30,980 --> 00:27:33,070
So why is it in an
indeterminate form.
449
00:27:33,070 --> 00:27:38,600
This one goes to 0, and this
one goes to minus infinity.
450
00:27:38,600 --> 00:27:40,460
So, excuse me,
this is a product.
451
00:27:40,460 --> 00:27:45,999
It's 0 times minus infinity.
452
00:27:45,999 --> 00:27:47,790
So that's an indeterminate
form, because we
453
00:27:47,790 --> 00:27:49,730
don't know whether the 0
wins or the infinity this
454
00:27:49,730 --> 00:27:51,620
could keep getting smaller
and smaller and smaller,
455
00:27:51,620 --> 00:27:52,890
and this could be getting
bigger and bigger bigger.
456
00:27:52,890 --> 00:27:55,420
The product could be
anything in between.
457
00:27:55,420 --> 00:27:57,740
We just don't know.
458
00:27:57,740 --> 00:28:04,270
So the first step is to write
this as a ratio of things,
459
00:28:04,270 --> 00:28:06,840
rather than a product of things.
460
00:28:06,840 --> 00:28:08,450
And it turns out
that the way to do
461
00:28:08,450 --> 00:28:11,590
that is to use the
logarithm in the numerator,
462
00:28:11,590 --> 00:28:14,360
and the 1 / x in
the denominator.
463
00:28:14,360 --> 00:28:18,040
So this is a choice
that I'm making here.
464
00:28:18,040 --> 00:28:23,800
Now, I've just converted it
to a limit of the type minus
465
00:28:23,800 --> 00:28:28,100
infinity divided by infinity.
466
00:28:28,100 --> 00:28:30,760
Because the numerator is going
to minus infinity as x goes
467
00:28:30,760 --> 00:28:37,550
to 0+ and the denominator 1 /
x is going to plus infinity.
468
00:28:37,550 --> 00:28:40,300
Again, there's a competition,
but now it's one of the forms
469
00:28:40,300 --> 00:28:44,210
to which L'Hôpital's
Rule applies.
470
00:28:44,210 --> 00:28:49,400
Now I'm just going to
apply L'Hôpital's Rule.
471
00:28:49,400 --> 00:28:54,251
And what it says is that
I differentiate here.
472
00:28:54,251 --> 00:28:56,500
So I just differentiate the
numerator and denominator.
473
00:28:56,500 --> 00:28:58,900
Applying L'Hôpital's
Rule is a breeze.
474
00:28:58,900 --> 00:29:03,710
You just differentiate,
differentiate.
475
00:29:03,710 --> 00:29:06,810
And now it just
simplifies and we're done.
476
00:29:06,810 --> 00:29:14,180
This is the limit as x goes to
0+ of, well, the x^2's cancel.
477
00:29:14,180 --> 00:29:20,360
This is the same as just
-x. x factors cancel.
478
00:29:20,360 --> 00:29:21,820
And so that's 0.
479
00:29:21,820 --> 00:29:24,170
The answer is that it's 0.
480
00:29:24,170 --> 00:29:30,650
So x goes to 0 faster then
ln n goes to minus infinity.
481
00:29:30,650 --> 00:29:36,400
This 0 was the winner.
482
00:29:36,400 --> 00:29:44,240
Something you can't
necessarily predict in advance.
483
00:29:44,240 --> 00:29:49,920
So let's do the other two
examples that I wrote down.
484
00:29:49,920 --> 00:29:53,230
I'm going to do them in
slightly more generality,
485
00:29:53,230 --> 00:29:57,480
because they're the
most fundamental rate
486
00:29:57,480 --> 00:29:59,370
properties that
you're going to need
487
00:29:59,370 --> 00:30:01,220
to know for the next section.
488
00:30:01,220 --> 00:30:03,000
Which is improper integrals.
489
00:30:03,000 --> 00:30:07,390
And also they're just very
important for physical math,
490
00:30:07,390 --> 00:30:10,390
and any other kind
of thing, basically.
491
00:30:10,390 --> 00:30:12,340
So here, let's just do these.
492
00:30:12,340 --> 00:30:16,210
So let's see, which one
do I want to do first.
493
00:30:16,210 --> 00:30:21,179
So I wrote down the
limit of x e^(-x),
494
00:30:21,179 --> 00:30:22,970
but I'm going to make
it even more general.
495
00:30:22,970 --> 00:30:25,830
I'm going to make it
any negative power
496
00:30:25,830 --> 00:30:30,190
here, where p is some
positive constant.
497
00:30:30,190 --> 00:30:35,680
Now again, this is a product
of functions, not a quotient,
498
00:30:35,680 --> 00:30:37,350
a ratio, of functions.
499
00:30:37,350 --> 00:30:41,010
And so I need to rewrite it.
500
00:30:41,010 --> 00:30:50,670
I'm going to write
it as x / e^(px). p
501
00:30:50,670 --> 00:30:52,080
And now I'm going
to apply, well,
502
00:30:52,080 --> 00:30:58,420
so it's of this form
infinity / infinity.
503
00:30:58,420 --> 00:31:01,860
And now that's the same as the
limit as x goes to infinity
504
00:31:01,860 --> 00:31:04,080
of 1 / (p e^(px)).
505
00:31:07,520 --> 00:31:08,670
So where does that go?
506
00:31:08,670 --> 00:31:10,250
As x goes to infinity.
507
00:31:10,250 --> 00:31:12,710
Now we can decide.
508
00:31:12,710 --> 00:31:14,430
The 1 stays where it is.
509
00:31:14,430 --> 00:31:23,440
And this, as x goes to
infinity, goes to infinity.
510
00:31:23,440 --> 00:31:27,080
So the answer is 0.
511
00:31:27,080 --> 00:31:48,184
And the conclusion is that x
grows more slowly then e^(px).
512
00:31:48,184 --> 00:31:49,100
As x goes to infinity.
513
00:31:49,100 --> 00:31:50,980
Remember, p is positive
here, of course.
514
00:31:50,980 --> 00:31:53,990
It's the increasing
exponentials.
515
00:31:53,990 --> 00:32:03,080
Not the decreasing ones.
516
00:32:03,080 --> 00:32:08,530
Let's do a variant of this.
517
00:32:08,530 --> 00:32:10,690
I'll do it the opposite way.
518
00:32:10,690 --> 00:32:13,580
So I'm going to call
this Example 5'.
519
00:32:13,580 --> 00:32:15,550
It really doesn't give
us any more information,
520
00:32:15,550 --> 00:32:18,280
but it gives you just a
little bit more practice.
521
00:32:18,280 --> 00:32:26,170
So suppose I look at
things the other way.
522
00:32:26,170 --> 00:32:35,260
e^(px) divided by, say, x^100.
523
00:32:35,260 --> 00:32:42,010
Now, this is an infinity
/ infinity example, again.
524
00:32:42,010 --> 00:32:44,820
And you can work
out what it's doing.
525
00:32:44,820 --> 00:32:47,897
But there are two ways
of thinking about this.
526
00:32:47,897 --> 00:32:49,480
There's the slow way
and the fast way.
527
00:32:49,480 --> 00:32:54,390
The slow way is to
differentiate this 100 times.
528
00:32:54,390 --> 00:32:55,530
That is, right?
529
00:32:55,530 --> 00:32:58,490
Apply L'Hôpital's Rule over and
over and over and over again.
530
00:32:58,490 --> 00:33:00,430
All the way.
531
00:33:00,430 --> 00:33:03,570
It's clear that you could do
it, but it's kind of a nuisance.
532
00:33:03,570 --> 00:33:06,850
So there's a much
cleverer trick here.
533
00:33:06,850 --> 00:33:12,650
Which is to change this to the
limit, as x goes to infinity,
534
00:33:12,650 --> 00:33:19,610
of the e ^ e^(px/100) /
x, to the 100th power.
535
00:33:25,850 --> 00:33:32,590
So if you do that, then we just
have one L'Hôpital's Rule step
536
00:33:32,590 --> 00:33:34,540
here.
537
00:33:34,540 --> 00:33:43,170
And that one is that this
is the same as, ...as x goes
538
00:33:43,170 --> 00:33:51,110
to infinity of, well it's
p/100 e^(px/100) divided by 1,
539
00:33:51,110 --> 00:33:53,270
all to the 100th power.
540
00:33:55,830 --> 00:34:02,480
That's our L'Hôpital step.
541
00:34:02,480 --> 00:34:09,310
And of course, that's
(infinity / 1)^100.
542
00:34:09,310 --> 00:34:10,410
Which is infinity.
543
00:34:10,410 --> 00:34:13,920
Now, again I did this in
a slightly different way
544
00:34:13,920 --> 00:34:16,800
to show you that it works
with infinity as well.
545
00:34:16,800 --> 00:34:18,400
So that was this other case.
546
00:34:18,400 --> 00:34:20,730
The right-hand side
can exist, or it
547
00:34:20,730 --> 00:34:22,460
can be plus or minus infinity.
548
00:34:22,460 --> 00:34:25,240
And that applies to this limit.
549
00:34:25,240 --> 00:34:27,990
And therefore, to
the original limit.
550
00:34:27,990 --> 00:34:35,180
And the conclusion here
is that e^(px), p > 0,
551
00:34:35,180 --> 00:34:46,340
grows faster than
any power of x.
552
00:34:46,340 --> 00:34:50,290
I picked x^100, but obviously
it didn't matter what power I
553
00:34:50,290 --> 00:34:52,670
picked.
554
00:34:52,670 --> 00:35:02,050
The exponents beat
all the powers.
555
00:35:02,050 --> 00:35:05,120
So we have one more of the ones
that I gave at the beginning
556
00:35:05,120 --> 00:35:07,780
to take care of.
557
00:35:07,780 --> 00:35:11,330
And that one is the logarithm.
558
00:35:11,330 --> 00:35:15,720
And its behavior at infinity.
559
00:35:15,720 --> 00:35:18,700
So I'll do a slight
variant on that one, too.
560
00:35:18,700 --> 00:35:24,070
So we have Example
6, which is ln x,
561
00:35:24,070 --> 00:35:27,780
and instead of dividing by x,
I'm going to divide by x^(1/3).
562
00:35:27,780 --> 00:35:29,990
I could divide by any
positive power of x,
563
00:35:29,990 --> 00:35:32,170
we'll just do this example here.
564
00:35:32,170 --> 00:35:37,080
So now this, as x
goes to infinity,
565
00:35:37,080 --> 00:35:43,230
is of the form
infinity / infinity.
566
00:35:43,230 --> 00:35:46,470
And so it's equivalent
to what happens
567
00:35:46,470 --> 00:35:49,250
when I differentiate numerator
and denominator separately.
568
00:35:49,250 --> 00:36:00,320
And that's 1 / x, and
here I have 1/3 x^(-2/3).
569
00:36:00,320 --> 00:36:03,470
1 / x, and then 1/3 x^(-2/3).
570
00:36:03,470 --> 00:36:06,600
Now, when the dust settles
here and you get your exponents
571
00:36:06,600 --> 00:36:10,560
right, we have an x^(-1),
and this is an x x^(-2/3),
572
00:36:10,560 --> 00:36:12,500
and that's a 1/3 becomes a 3.
573
00:36:12,500 --> 00:36:19,260
So this is what it is.
574
00:36:19,260 --> 00:36:23,000
And that's equal to 3x^(-1/3).
575
00:36:26,480 --> 00:36:27,830
Which we can decide.
576
00:36:27,830 --> 00:36:30,150
It goes to 0.
577
00:36:30,150 --> 00:36:37,240
As x goes to infinity.
578
00:36:37,240 --> 00:36:50,200
And so the conclusion is that
ln x grows more slowly as x goes
579
00:36:50,200 --> 00:37:08,690
to infinity, than x x^(1/3)
or any positive power of x.
580
00:37:08,690 --> 00:37:15,500
So any x^p, p
positive, will work.
581
00:37:15,500 --> 00:37:17,670
So log is really slow,
going to infinity.
582
00:37:17,670 --> 00:37:20,550
It's very, very gradual.
583
00:37:20,550 --> 00:37:21,420
Yeah, question.
584
00:37:21,420 --> 00:37:45,970
STUDENT: [INAUDIBLE]
585
00:37:45,970 --> 00:37:48,230
PROFESSOR: The
question is, how many
586
00:37:48,230 --> 00:37:50,630
hypotheses do you need here?
587
00:37:50,630 --> 00:37:57,590
So I said that, and I think
what you were asking is,
588
00:37:57,590 --> 00:38:02,940
if I have this hypothesis, can
I also have this hypothesis.
589
00:38:02,940 --> 00:38:04,870
That's OK.
590
00:38:04,870 --> 00:38:08,840
I can have this hypothesis
combined with this one.
591
00:38:08,840 --> 00:38:11,280
I need something
about f(a) and g(a).
592
00:38:11,280 --> 00:38:14,910
I can't assume nothing
about f(a) and g(a).
593
00:38:14,910 --> 00:38:18,190
So in other words, I have to be
faced with either an infinity /
594
00:38:18,190 --> 00:38:24,180
infinity, or a 0 / 0 situation.
595
00:38:24,180 --> 00:38:26,650
So let's see.
596
00:38:26,650 --> 00:38:35,160
The rule applies in the 0 / 0,
or infinity / infinity case.
597
00:38:35,160 --> 00:38:40,910
These are the only two
cases that it applies in.
598
00:38:40,910 --> 00:38:45,370
And a can be anything.
599
00:38:45,370 --> 00:38:48,880
Including infinity.
600
00:38:48,880 --> 00:38:51,230
Plus or minus infinity.
601
00:38:51,230 --> 00:38:53,390
The rule applies
in these two cases.
602
00:38:53,390 --> 00:38:58,310
So in other words, this
is what f(a) / g(a) is.
603
00:38:58,310 --> 00:39:00,750
Either one of these.
604
00:39:00,750 --> 00:39:02,540
And in fact, it can
be plus or minus.
605
00:39:02,540 --> 00:39:06,460
STUDENT: [INAUDIBLE]
606
00:39:06,460 --> 00:39:10,290
PROFESSOR: And the right-hand
side has to be something.
607
00:39:10,290 --> 00:39:21,250
It has to be either finite
or plus or minus infinity.
608
00:39:21,250 --> 00:39:23,690
So you need something.
609
00:39:23,690 --> 00:39:25,680
You need a specific
value of a, you
610
00:39:25,680 --> 00:39:28,070
need to decide whether
it's an indeterminate form.
611
00:39:28,070 --> 00:39:30,320
And you need the
right-hand limit to exist.
612
00:39:30,320 --> 00:39:33,530
It's not hard to impose this.
613
00:39:33,530 --> 00:39:35,975
Because when you look
at the right-hand side,
614
00:39:35,975 --> 00:39:37,350
you'll want to be
calculating it.
615
00:39:37,350 --> 00:39:38,641
So you want to know what it is.
616
00:39:38,641 --> 00:39:47,640
So you'll never have problems
confirming this hypothesis.
617
00:39:47,640 --> 00:39:51,280
Alright.
618
00:39:51,280 --> 00:39:54,480
Let me give you one
more example here.
619
00:39:54,480 --> 00:39:56,600
Which is just slightly trickier.
620
00:39:56,600 --> 00:40:15,130
Which involves, so here's
another indeterminate form.
621
00:40:15,130 --> 00:40:16,780
That's going to be 0^0.
622
00:40:20,351 --> 00:40:22,350
So there are lots of these
things where you just
623
00:40:22,350 --> 00:40:23,690
don't know what to do.
624
00:40:23,690 --> 00:40:27,730
And they come out in
various different ways.
625
00:40:27,730 --> 00:40:32,620
The simplest example of this is
the limit as x goes to 0 from
626
00:40:32,620 --> 00:40:34,280
above of x^x.
627
00:40:41,430 --> 00:40:45,120
In order to work out what's
happening with this one,
628
00:40:45,120 --> 00:40:47,610
we have to use a trick.
629
00:40:47,610 --> 00:40:52,650
And the trick is this
is a moving exponent.
630
00:40:52,650 --> 00:40:56,220
And so it's appropriate
to use base e.
631
00:40:56,220 --> 00:40:59,010
This is something that we did
way back in the first unit.
632
00:40:59,010 --> 00:41:06,690
So, since we have
a moving exponent,
633
00:41:06,690 --> 00:41:11,830
we're going to use base e.
634
00:41:11,830 --> 00:41:13,600
That's the good
base to use whenever
635
00:41:13,600 --> 00:41:15,600
you have a moving exponent.
636
00:41:15,600 --> 00:41:18,570
And so rewrite this
as x^x = e^(x ln x).
637
00:41:21,530 --> 00:41:23,700
And now, in order to figure
out what's happening,
638
00:41:23,700 --> 00:41:25,250
we really only
have to know what's
639
00:41:25,250 --> 00:41:32,140
going on with the exponent.
640
00:41:32,140 --> 00:41:34,100
So remember, actually
we already did this.
641
00:41:34,100 --> 00:41:36,100
But I'm going to do
it once more for you.
642
00:41:36,100 --> 00:41:39,340
This is ln x / (1/x).
643
00:41:39,340 --> 00:41:42,090
And that's equivalent,
as x goes to 0,
644
00:41:42,090 --> 00:41:50,690
to using L'Hôpital's Rule
to 1/x, and this is -1/x^2,
645
00:41:50,690 --> 00:41:54,350
which is -x, which goes to 0.
646
00:41:54,350 --> 00:41:58,030
As x goes to 0.
647
00:41:58,030 --> 00:42:01,370
And so what we have here
is that this one is going
648
00:42:01,370 --> 00:42:05,140
to be equivalent
to, well, it's going
649
00:42:05,140 --> 00:42:07,520
to tend to what
we got over here.
650
00:42:07,520 --> 00:42:10,170
It's e^0.
651
00:42:10,170 --> 00:42:13,980
That exponent is what we want.
652
00:42:13,980 --> 00:42:18,250
As x goes to 0.
653
00:42:18,250 --> 00:42:27,190
So that's the answer This
limit happens to be 1.
654
00:42:27,190 --> 00:42:28,940
That's actually
relatively easy to do,
655
00:42:28,940 --> 00:42:42,700
given all of the power
that we have at our hands.
656
00:42:42,700 --> 00:42:49,310
Now, let me give you
one more example.
657
00:42:49,310 --> 00:42:54,280
Suppose you're trying to
understand the limit of sin x /
658
00:42:54,280 --> 00:42:54,780
x^2.
659
00:42:59,060 --> 00:43:06,090
If you apply L'Hôpital's
Rule, as x goes to 0,
660
00:43:06,090 --> 00:43:11,660
you're going to
get cos x / (2x).
661
00:43:11,660 --> 00:43:19,270
And if you apply L'Hôpital's
Rule again, as x goes to 0,
662
00:43:19,270 --> 00:43:24,920
you're going to get - sin x / 2.
663
00:43:24,920 --> 00:43:35,840
And this, as x goes
to 0, goes to 0.
664
00:43:35,840 --> 00:43:39,820
On the other hand, if you look
at the linear approximation
665
00:43:39,820 --> 00:43:49,470
method, linear approximation
says that sin x
666
00:43:49,470 --> 00:43:55,090
is approximately x near 0.
667
00:43:55,090 --> 00:43:59,560
So that should be x / x^2.
668
00:43:59,560 --> 00:44:04,900
Which is 1 / x, which
goes to infinity.
669
00:44:04,900 --> 00:44:08,140
As x goes to 0, at
least from one side,
670
00:44:08,140 --> 00:44:13,100
minus infinity to
the other side.
671
00:44:13,100 --> 00:44:17,320
So there's something fishy
going on here, right?
672
00:44:17,320 --> 00:44:19,330
So this is fishy.
673
00:44:19,330 --> 00:44:21,520
Or maybe this is
fishy, I don't know.
674
00:44:21,520 --> 00:44:26,251
So, tell me what's wrong here.
675
00:44:26,251 --> 00:44:26,750
Yeah.
676
00:44:26,750 --> 00:44:38,230
STUDENT: [INAUDIBLE]
PROFESSOR: OK.
677
00:44:38,230 --> 00:44:42,615
So the claim is that
the second application
678
00:44:42,615 --> 00:44:51,950
of L'Hôpital's Rule,
this one, is wrong.
679
00:44:51,950 --> 00:44:54,620
And that's correct.
680
00:44:54,620 --> 00:44:56,700
And this is where you
have to watch out,
681
00:44:56,700 --> 00:44:58,087
with L'Hôpital's Rule.
682
00:44:58,087 --> 00:44:59,920
This is exactly where
you have to watch out.
683
00:44:59,920 --> 00:45:02,300
You have to apply the test.
684
00:45:02,300 --> 00:45:03,820
Here it's an indeterminate form.
685
00:45:03,820 --> 00:45:08,260
It's 0 / 0 before
I applied the rule.
686
00:45:08,260 --> 00:45:10,630
But in order to apply
the rule the second time,
687
00:45:10,630 --> 00:45:12,410
it still has to be 0 / 0.
688
00:45:12,410 --> 00:45:14,170
But this one isn't.
689
00:45:14,170 --> 00:45:19,109
This one is 1 / 0.
690
00:45:19,109 --> 00:45:20,650
It's no longer an
indeterminate form.
691
00:45:20,650 --> 00:45:22,895
It's actually infinite.
692
00:45:22,895 --> 00:45:25,520
Either plus or minus, depending
on the sign of the denominator.
693
00:45:25,520 --> 00:45:27,970
Which is just what
this answer is.
694
00:45:27,970 --> 00:45:30,550
So the linear
approximation is safe.
695
00:45:30,550 --> 00:45:35,160
And we just applied
L'Hôpital's Rule wrong.
696
00:45:35,160 --> 00:45:55,760
So the moral of the story
here is look before you L'Hôp.
697
00:45:55,760 --> 00:45:58,260
Alright.
698
00:45:58,260 --> 00:46:09,850
Now, let me say one more thing.
699
00:46:09,850 --> 00:46:22,900
I need to pile it on
just a little bit, sorry.
700
00:46:22,900 --> 00:46:36,700
So don't use it as a crutch.
701
00:46:36,700 --> 00:46:39,110
We don't want to just
get ourselves so weak,
702
00:46:39,110 --> 00:46:41,190
after being in the
hospital for all this time,
703
00:46:41,190 --> 00:46:55,560
that we can't use, I'm sorry.
704
00:46:55,560 --> 00:47:00,700
So remember that you shouldn't
have lost your senses.
705
00:47:00,700 --> 00:47:12,210
If you have something like
this, so we'll do this one here.
706
00:47:12,210 --> 00:47:15,030
Suppose you're trying
to understand what this
707
00:47:15,030 --> 00:47:18,200
does as x goes to infinity.
708
00:47:18,200 --> 00:47:25,700
Now, you could apply L'Hôpital's
Rule five times, or four times.
709
00:47:25,700 --> 00:47:30,040
And get the answer here.
710
00:47:30,040 --> 00:47:33,570
But really, you should realize
that the main terms are sitting
711
00:47:33,570 --> 00:47:34,870
there right in front of you.
712
00:47:34,870 --> 00:47:36,453
And that there's
some algebra that you
713
00:47:36,453 --> 00:47:38,490
can do to simplify this.
714
00:47:38,490 --> 00:47:44,480
Namely, it's the same
as 1 + 2/x + 1/x^5.
715
00:47:48,060 --> 00:47:51,510
And then in the denominator,
well, let's see.
716
00:47:51,510 --> 00:47:53,110
It's x.
717
00:47:53,110 --> 00:47:57,574
So this would be dividing
by 1/x^5 in both numerator
718
00:47:57,574 --> 00:47:58,240
and denominator.
719
00:47:58,240 --> 00:48:04,410
And here you have 1/x plus
2 over, sorry I overshot.
720
00:48:04,410 --> 00:48:06,830
But that's OK.
721
00:48:06,830 --> 00:48:09,110
2/x^5 here.
722
00:48:09,110 --> 00:48:12,650
So these are the main
terms, if you like.
723
00:48:12,650 --> 00:48:18,510
And it's the same as 1 /
(1/x), which is the same as x,
724
00:48:18,510 --> 00:48:21,490
and it goes to infinity.
725
00:48:21,490 --> 00:48:22,930
As x goes to infinity.
726
00:48:22,930 --> 00:48:25,130
Or, if you like,
much more simply,
727
00:48:25,130 --> 00:48:29,230
just x^5 / x^4 is the main term.
728
00:48:29,230 --> 00:48:30,520
Which is x.
729
00:48:30,520 --> 00:48:31,590
Which goes to infinity.
730
00:48:31,590 --> 00:48:35,350
So don't forget
your basic algebra
731
00:48:35,350 --> 00:48:37,210
when you're doing
this kind of stuff.
732
00:48:37,210 --> 00:48:40,640
Use these things and don't
use L'Hôpital's Rule.
733
00:48:40,640 --> 00:48:42,355
OK, see you next time.