1
00:00:00,000 --> 00:00:00,730


2
00:00:00,730 --> 00:00:06,839
In this video I will prove to you that the limit as

3
00:00:06,839 --> 00:00:15,289
x approaches 0 of sine of x over x is equal to 1.

4
00:00:15,289 --> 00:00:18,530
But before I do that, before I break into trigonometry, I'm

5
00:00:18,530 --> 00:00:22,650
going to go over another aspect of limits.

6
00:00:22,649 --> 00:00:24,139
And that's the squeeze theorem.

7
00:00:24,140 --> 00:00:26,199
Because once you understand what the squeeze theorem is,

8
00:00:26,199 --> 00:00:30,149
we can use the squeeze theorem to prove this.

9
00:00:30,149 --> 00:00:33,509
It's actually a pretty involved explanation, but I think you'll

10
00:00:33,509 --> 00:00:37,159
find it pretty neat and satisfying if you get it.

11
00:00:37,159 --> 00:00:39,219
If you don't get it, maybe you just want to memorize this.

12
00:00:39,219 --> 00:00:41,579
Because that's a very useful limit to know later on when

13
00:00:41,579 --> 00:00:43,679
we take the derivatives of trig functions.

14
00:00:43,679 --> 00:00:45,240
So what's the squeeze theorem?

15
00:00:45,240 --> 00:00:50,039
The squeeze theorem is my favorite theorem in

16
00:00:50,039 --> 00:00:53,789
mathematics, possibly because it has the word squeeze in it.

17
00:00:53,789 --> 00:00:56,560
Squeeze theorem.

18
00:00:56,560 --> 00:00:58,310
And when you read it in a calculus book it looks

19
00:00:58,310 --> 00:00:59,740
all complicated.

20
00:00:59,740 --> 00:01:01,580
I don't know when you read it, in a calculus book or

21
00:01:01,579 --> 00:01:02,500
in a precalculus book.

22
00:01:02,500 --> 00:01:05,079
It looks all complicated, but what it's saying is

23
00:01:05,079 --> 00:01:07,439
frankly pretty obvious.

24
00:01:07,439 --> 00:01:08,409
Let me give you an example.

25
00:01:08,409 --> 00:01:16,709
If I told you that I always-- so Sal always

26
00:01:16,709 --> 00:01:23,149
eats more than Umama.

27
00:01:23,150 --> 00:01:25,650
Umama is my wife.

28
00:01:25,650 --> 00:01:27,670
If I told you that this is true, Sal always

29
00:01:27,670 --> 00:01:29,370
eats more than Umama.

30
00:01:29,370 --> 00:01:42,990
And I were also to say that Sal always eats less than-- I don't

31
00:01:42,989 --> 00:01:45,189
know, let me make up a fictional character--

32
00:01:45,189 --> 00:01:45,834
than Bill.

33
00:01:45,834 --> 00:01:48,369


34
00:01:48,370 --> 00:01:52,020
So on any given day-- let's say this is in a given day.

35
00:01:52,019 --> 00:01:58,119
Sal always eats more than Umama in any given day, and Sal

36
00:01:58,120 --> 00:02:01,600
always eats less than Bill on any given day.

37
00:02:01,599 --> 00:02:15,359
Now if I were tell you that on Tuesday Umama ate 300 calories

38
00:02:15,360 --> 00:02:18,840
and on Tuesday Bill ate 300 calories.

39
00:02:18,840 --> 00:02:21,390


40
00:02:21,389 --> 00:02:25,819
So my question to you is, how many calories did Sal eat,

41
00:02:25,819 --> 00:02:28,150
or did I eat, on Tuesday?

42
00:02:28,150 --> 00:02:33,379
Well I always eat more than Umama-- well, more than or

43
00:02:33,379 --> 00:02:37,299
equal to Umama-- and I always eat less than or equal to Bill.

44
00:02:37,300 --> 00:02:41,350
So then on Tuesday, I must have eaten 300 calories.

45
00:02:41,349 --> 00:02:43,949
So this is the gist of the squeeze theorem, and I'll do

46
00:02:43,949 --> 00:02:44,939
a little bit more formally.

47
00:02:44,939 --> 00:02:48,710
But it's essentially saying, if I'm always greater than one

48
00:02:48,710 --> 00:02:52,189
thing and I'm always less than another thing and at some point

49
00:02:52,189 --> 00:02:55,560
those two things are equal, well then I must be equal

50
00:02:55,560 --> 00:02:57,120
to whatever those two things are equal to.

51
00:02:57,120 --> 00:02:59,080
I've kind of been squeezed in between them.

52
00:02:59,080 --> 00:03:01,600
I'm always in between Umama and Bill, and if they're at the

53
00:03:01,599 --> 00:03:04,219
exact same point on Tuesday, then I must be at

54
00:03:04,219 --> 00:03:05,000
that point as well.

55
00:03:05,000 --> 00:03:06,360
Or at least I must approach it.

56
00:03:06,360 --> 00:03:08,290
So let me write it in math terms.

57
00:03:08,289 --> 00:03:11,879


58
00:03:11,879 --> 00:03:18,729
So all it says is that, over some domain, if I say that,

59
00:03:18,729 --> 00:03:25,299
let's say that g of x is less than or equal to f of x, which

60
00:03:25,300 --> 00:03:29,310
is less than or equal to h of x over some domain.

61
00:03:29,310 --> 00:03:38,719
And we also know that the limit of g of x as x approaches a is

62
00:03:38,719 --> 00:03:45,069
equal to some limit, capital L, and we also know that the limit

63
00:03:45,069 --> 00:03:52,139
as x approaches a of h of x also equals L, then the squeeze

64
00:03:52,139 --> 00:03:55,199
theorem tells us-- and I'm not going to prove that right

65
00:03:55,199 --> 00:03:57,539
here, but it's good to just understand what the squeeze

66
00:03:57,539 --> 00:04:02,699
theorem is-- the squeeze theorem tells us then the limit

67
00:04:02,699 --> 00:04:09,769
as x approaches a of f of x must also be equal to L.

68
00:04:09,770 --> 00:04:11,230
And this is the same thing.

69
00:04:11,229 --> 00:04:14,090
This is example where f of x, this could be how much Sal eats

70
00:04:14,090 --> 00:04:16,410
in a day, this could be how much Umama eats in a

71
00:04:16,410 --> 00:04:17,330
day, this is Bill.

72
00:04:17,329 --> 00:04:19,979
So I always eat more than Umama or less than Bill.

73
00:04:19,980 --> 00:04:25,189
And then on Tuesday, you could say a is Tuesday, if Umama had

74
00:04:25,189 --> 00:04:28,649
300 calories and Bill had 300 calories, then I also had

75
00:04:28,649 --> 00:04:29,479
to eat 300 calories.

76
00:04:29,480 --> 00:04:32,350
Let me let me graph that for you.

77
00:04:32,350 --> 00:04:36,470
Let me graph that, and I'll do it in a different color.

78
00:04:36,470 --> 00:04:37,790
Squeeze theorem.

79
00:04:37,790 --> 00:04:42,560


80
00:04:42,560 --> 00:04:44,050
Squeeze theorem.

81
00:04:44,050 --> 00:04:51,942
OK, so let's draw the point a comma L.

82
00:04:51,942 --> 00:04:53,980
The point a comma L.

83
00:04:53,980 --> 00:04:55,840
Let's say this is a, that's the point that we care

84
00:04:55,839 --> 00:04:59,899
about. a, and this is L.

85
00:04:59,899 --> 00:05:03,769
And we know, g of x, that's the lower function, right?

86
00:05:03,769 --> 00:05:05,539
So let's say that this green thing right

87
00:05:05,540 --> 00:05:07,540
here, this is g of x.

88
00:05:07,540 --> 00:05:10,030
So this is my g of x.

89
00:05:10,029 --> 00:05:14,109
And we know that as g of x approaches-- so the g of x

90
00:05:14,110 --> 00:05:16,120
could look something like that, right?

91
00:05:16,120 --> 00:05:18,910
And we know that the limit as x approaches a of

92
00:05:18,910 --> 00:05:21,510
g of x is equal to L.

93
00:05:21,509 --> 00:05:23,589
So that's right there.

94
00:05:23,589 --> 00:05:26,859
So this is g of x.

95
00:05:26,860 --> 00:05:28,509
That's g of x.

96
00:05:28,509 --> 00:05:31,589
Let me do h of x in a different color.

97
00:05:31,589 --> 00:05:33,569
So now h of x could look something like this.

98
00:05:33,569 --> 00:05:36,790


99
00:05:36,790 --> 00:05:38,740
Like that.

100
00:05:38,740 --> 00:05:41,870
So that's h of x.

101
00:05:41,870 --> 00:05:45,970
And we also know that the limit as x approaches a of h of x --

102
00:05:45,970 --> 00:05:51,610
let's see, this is the function of x axis.

103
00:05:51,610 --> 00:05:56,629
So you can call it h of x, g of x, or f of x.

104
00:05:56,629 --> 00:06:00,350
That's just the dependent access, and this is the x-axis.

105
00:06:00,350 --> 00:06:04,790
So once again, the limit as x approaches a of h of x, well

106
00:06:04,790 --> 00:06:07,569
at that point right there, h of a is equal to L.

107
00:06:07,569 --> 00:06:08,964
Or at least the limit is equal to that.

108
00:06:08,964 --> 00:06:11,469


109
00:06:11,470 --> 00:06:13,530
And none of these functions actually have to even be

110
00:06:13,529 --> 00:06:17,209
defined at a, as long as these limits, this limit exists

111
00:06:17,209 --> 00:06:18,109
and this limit exists.

112
00:06:18,110 --> 00:06:20,790
And that's also an important thing to keep in mind.

113
00:06:20,790 --> 00:06:23,730
So what does this tell us? f of x is always greater

114
00:06:23,730 --> 00:06:24,860
than this green function.

115
00:06:24,860 --> 00:06:27,350
It's always less than h of x, right?

116
00:06:27,350 --> 00:06:29,910
So any f of x I draw, it would have to be in

117
00:06:29,910 --> 00:06:31,080
between those two, right?

118
00:06:31,079 --> 00:06:34,800
So no matter how I draw it, if I were to draw a function,

119
00:06:34,800 --> 00:06:38,600
it's bounded by those two functions just by definition.

120
00:06:38,600 --> 00:06:40,500
So it has to go through that point.

121
00:06:40,500 --> 00:06:41,910
Or at least it has to approach that point.

122
00:06:41,910 --> 00:06:45,060
Maybe it's not defined at that point, but the limit as we

123
00:06:45,060 --> 00:06:49,949
approach a of f of x also has to be at point L.

124
00:06:49,949 --> 00:06:52,579
And maybe f of x doesn't have to be defined right there, but

125
00:06:52,579 --> 00:06:54,939
the limit as we approach it is going to be L.

126
00:06:54,939 --> 00:06:56,920
And hopefully that makes a little bit of sense, and

127
00:06:56,920 --> 00:06:59,250
hopefully my calories example made a little

128
00:06:59,250 --> 00:06:59,819
bit of sense to you.

129
00:06:59,819 --> 00:07:01,969
So let's keep that in the back of our mind,

130
00:07:01,970 --> 00:07:03,860
the squeeze theorem.

131
00:07:03,860 --> 00:07:11,970
And now we will use that to prove that the limit as x

132
00:07:11,970 --> 00:07:16,310
approaches 0 of sine of x over x is equal to 1.

133
00:07:16,310 --> 00:07:17,939
And I want to do that, one, because this is

134
00:07:17,939 --> 00:07:18,990
a super useful limit.

135
00:07:18,990 --> 00:07:20,600
And then the other thing is, sometimes you learn the squeeze

136
00:07:20,600 --> 00:07:22,770
theorem, you're like, oh, well that's obvious but

137
00:07:22,769 --> 00:07:23,889
when is it useful?

138
00:07:23,889 --> 00:07:25,089
And we'll see.

139
00:07:25,089 --> 00:07:26,519
Actually I'm going to do it in the next video, since we're

140
00:07:26,519 --> 00:07:27,589
already pushing 8 minutes.

141
00:07:27,589 --> 00:07:29,339
But we'll see in the next video that the squeeze theorem is

142
00:07:29,339 --> 00:07:32,500
tremendously useful when we're trying to prove this.

143
00:07:32,500 --> 00:07:35,079
I will see you in the next video.

144
00:07:35,079 --> 00:07:36,399