1
00:00:00,000 --> 00:00:00,660


2
00:00:00,660 --> 00:00:08,369
Let's say we need to evaluate the limit as x approaches 0 of

3
00:00:08,369 --> 00:00:20,339
2 sine of x minus sine of 2x, all of that over x

4
00:00:20,339 --> 00:00:24,210
minus sine of x.

5
00:00:24,210 --> 00:00:26,820
Now, the first thing that I always try to do when I first

6
00:00:26,820 --> 00:00:29,160
see a limit problem is hey, what happens if I just try to

7
00:00:29,160 --> 00:00:33,890
evaluate this function at x is equal to 0?

8
00:00:33,890 --> 00:00:35,539
Maybe nothing crazy happens.

9
00:00:35,539 --> 00:00:37,089
So let's just try it out.

10
00:00:37,090 --> 00:00:40,340
If we try to do x equals 0, what happens?

11
00:00:40,340 --> 00:00:43,580
We get 2 sine of 0, which is 0.

12
00:00:43,579 --> 00:00:46,189
Minus sine of 2 times 0.

13
00:00:46,189 --> 00:00:49,960
Well, that's going to be sine of 0 again, which is 0.

14
00:00:49,960 --> 00:00:53,509
So our numerator is going to be equal to 0.

15
00:00:53,509 --> 00:00:54,750
Sine of 0, that's 0.

16
00:00:54,750 --> 00:00:56,439
And then we have another sine of 0 there.

17
00:00:56,439 --> 00:00:58,339
That's another 0, so all 0's.

18
00:00:58,340 --> 00:00:59,590
And our denominator, we're going to have

19
00:00:59,590 --> 00:01:02,590
a 0 minus sine of 0.

20
00:01:02,590 --> 00:01:05,200
Well that's also going to be 0.

21
00:01:05,200 --> 00:01:08,340
But we have that indeterminate form, we have that undefined

22
00:01:08,340 --> 00:01:12,000
0/0 that we talked about in the last video.

23
00:01:12,000 --> 00:01:15,719
So maybe we can use L'Hopital's rule here.

24
00:01:15,719 --> 00:01:18,909
In order to use L'Hopital's rule then the limit as x

25
00:01:18,909 --> 00:01:21,879
approaches 0 of the derivative of this function over

26
00:01:21,879 --> 00:01:24,879
the derivative of this function needs to exist.

27
00:01:24,879 --> 00:01:28,060
So let's just apply L'Hopital's rule and let's just take the

28
00:01:28,060 --> 00:01:30,469
derivative of each of these and see if we can find the limit.

29
00:01:30,469 --> 00:01:33,620
If we can, then that's going to be the limit of this thing.

30
00:01:33,620 --> 00:01:38,750
So this thing, assuming that it exists, is going to be equal to

31
00:01:38,750 --> 00:01:44,950
the limit as x approaches 0 of the derivative of this

32
00:01:44,950 --> 00:01:46,600
numerator up here.

33
00:01:46,599 --> 00:01:48,379
And so what's the derivative of the numerator going to be?

34
00:01:48,379 --> 00:01:49,314
I'll do it in a new color.

35
00:01:49,314 --> 00:01:51,049
I'll do it in green.

36
00:01:51,049 --> 00:01:55,370
Well, the derivative of 2 sine of x is 2 cosine of x.

37
00:01:55,370 --> 00:01:57,920


38
00:01:57,920 --> 00:02:00,159
And then, minus-- well, the derivative of sine

39
00:02:00,159 --> 00:02:04,149
of 2x is 2 cosine of 2x.

40
00:02:04,150 --> 00:02:07,440
So minus 2 cosine of 2x.

41
00:02:07,439 --> 00:02:09,210
Just use the chain rule there, derivative of

42
00:02:09,210 --> 00:02:11,000
the inside is just 2.

43
00:02:11,000 --> 00:02:12,060
That's the 2 out there.

44
00:02:12,060 --> 00:02:15,405
Derivative of the outside is cosine of 2x, and we had that

45
00:02:15,405 --> 00:02:16,789
negative number out there.

46
00:02:16,789 --> 00:02:19,609
So that's the derivative of our numerator, maria, and

47
00:02:19,610 --> 00:02:20,500
what is the Derivative.

48
00:02:20,500 --> 00:02:21,919
of our denominator?

49
00:02:21,919 --> 00:02:25,429
Well, derivative of x is just 1, and derivative of sine

50
00:02:25,430 --> 00:02:26,870
of x is just cosine of x.

51
00:02:26,870 --> 00:02:29,920
So 1 minus cosine of x.

52
00:02:29,919 --> 00:02:31,949
So let's try to evaluate this limit.

53
00:02:31,949 --> 00:02:33,149
What do we get?

54
00:02:33,150 --> 00:02:36,180
If we put a 0 up here we're going to get 2 times cosine

55
00:02:36,180 --> 00:02:39,610
of 0, which is 2-- let me write it like this.

56
00:02:39,610 --> 00:02:42,760
So this is 2 times cosine of 0, which is 1.

57
00:02:42,759 --> 00:02:48,039
So it's 2 minus 2 cosine of 2 times 0.

58
00:02:48,039 --> 00:02:49,479
Let me write it this way.

59
00:02:49,479 --> 00:02:50,579
Actually, let me just do it this way.

60
00:02:50,580 --> 00:02:53,150
If we just straight up evaluate the limit of the numerator and

61
00:02:53,150 --> 00:02:54,270
the denominator, what are we going to get?

62
00:02:54,270 --> 00:02:57,710
We get 2 cosine of 0, which is 2.

63
00:02:57,710 --> 00:03:01,760
Minus 2 times cosine of-- well, this 2 times 0 is

64
00:03:01,759 --> 00:03:02,620
still going to be 0.

65
00:03:02,620 --> 00:03:06,849
So minus 2 times cosine of 0, which is 2.

66
00:03:06,849 --> 00:03:15,620
All of that over 1 minus the cosine of 0, which is 1.

67
00:03:15,620 --> 00:03:17,295
So once again, we get 0/0.

68
00:03:17,294 --> 00:03:22,099


69
00:03:22,099 --> 00:03:24,500
So does this mean that the limit doesn't exist?

70
00:03:24,500 --> 00:03:27,419
No, it still might exist, we might just want to do

71
00:03:27,419 --> 00:03:28,829
L'Hopital's rule again.

72
00:03:28,830 --> 00:03:30,610
Let me take the derivative of that and put it over

73
00:03:30,610 --> 00:03:31,490
the derivative of that.

74
00:03:31,490 --> 00:03:34,610
And then take the limit and maybe L'Hopital's rule

75
00:03:34,610 --> 00:03:35,700
will help us on the next [INAUDIBLE].

76
00:03:35,699 --> 00:03:38,609
So let's see if it gets us anywhere.

77
00:03:38,610 --> 00:03:43,030
So this should be equal to the limit if L'Hopital's

78
00:03:43,030 --> 00:03:43,840
rule applies here.

79
00:03:43,840 --> 00:03:46,050
We're not 100% sure yet.

80
00:03:46,050 --> 00:03:49,950
This should be equal to the limit as x approaches 0 of the

81
00:03:49,949 --> 00:03:53,689
derivative of that thing over the derivative of that thing.

82
00:03:53,689 --> 00:03:58,240
So what's the derivative of 2 cosine of x?

83
00:03:58,240 --> 00:04:00,485
Well, derivative of cosine of x is negative sine of x.

84
00:04:00,485 --> 00:04:04,920
So it's negative 2 sine of x.

85
00:04:04,919 --> 00:04:11,319
And then derivative of cosine of 2x is negative 2 sine of 2x.

86
00:04:11,319 --> 00:04:13,659
So we're going to have this negative cancel out with the

87
00:04:13,659 --> 00:04:16,980
negative on the negative 2 and then a 2 times the 2.

88
00:04:16,980 --> 00:04:21,990
So it's going to be plus 4 sine of 2x.

89
00:04:21,990 --> 00:04:23,790
Let me make sure I did that right.

90
00:04:23,790 --> 00:04:26,640
We have the minus 2 or the negative 2 on the outside.

91
00:04:26,639 --> 00:04:30,949
Derivative of cosine of 2x is going to be 2 times

92
00:04:30,949 --> 00:04:32,589
negative sine of x.

93
00:04:32,589 --> 00:04:34,269
So the 2 times 2 is 4.

94
00:04:34,269 --> 00:04:36,779
The negative sine of x times-- the negative

95
00:04:36,779 --> 00:04:38,059
right there's a plus.

96
00:04:38,060 --> 00:04:40,180
You have a positive sine, so it's the sine of 2x.

97
00:04:40,180 --> 00:04:42,370
That's the numerator when you take the derivative.

98
00:04:42,370 --> 00:04:45,259
And the denominator-- this is just an exercise in

99
00:04:45,259 --> 00:04:45,829
taking derivatives.

100
00:04:45,829 --> 00:04:47,349
What's the derivative of the denominator?

101
00:04:47,350 --> 00:04:49,310
Derivative of 1 is 0.

102
00:04:49,310 --> 00:04:52,790
And derivative negative cosine of x is just-- well,

103
00:04:52,790 --> 00:04:53,900
that's just sine of x.

104
00:04:53,899 --> 00:04:56,479


105
00:04:56,480 --> 00:04:57,950
So let's take this limit.

106
00:04:57,949 --> 00:05:00,050
So this is going to be equal to-- well, immediately if I

107
00:05:00,050 --> 00:05:02,810
take x is equal to 0 in the denominator, I know that

108
00:05:02,810 --> 00:05:05,189
sine of 0 is just 0.

109
00:05:05,189 --> 00:05:06,870
Let's see what happens in the numerator.

110
00:05:06,870 --> 00:05:08,699
Negative 2 times sine of 0.

111
00:05:08,699 --> 00:05:10,740
That's going to be 0.

112
00:05:10,740 --> 00:05:13,879
And then plus 4 times sine of 2 times 0.

113
00:05:13,879 --> 00:05:16,569
Well, that's still sine of 0, so that's still going to be 0.

114
00:05:16,569 --> 00:05:19,209
So once again, we got indeterminate form again.

115
00:05:19,209 --> 00:05:19,899
Are we done?

116
00:05:19,899 --> 00:05:20,819
Do we give up?

117
00:05:20,819 --> 00:05:22,730
Do we say that L'Hopital's rule didn't work?

118
00:05:22,730 --> 00:05:26,550
No, because this could have been our first limit problem.

119
00:05:26,550 --> 00:05:28,750
And if this is our first limit problem we say, hey, maybe we

120
00:05:28,750 --> 00:05:30,680
could use L'Hopital's rule here because we got an

121
00:05:30,680 --> 00:05:32,600
indeterminate form.

122
00:05:32,600 --> 00:05:34,990
Both the numerator and the denominator approach

123
00:05:34,990 --> 00:05:36,660
0 as x approaches 0.

124
00:05:36,660 --> 00:05:40,110
So let's take the derivatives again.

125
00:05:40,110 --> 00:05:42,870
This will be equal to-- if the limit exist, the

126
00:05:42,870 --> 00:05:45,730
limit as x approaches 0.

127
00:05:45,730 --> 00:05:47,685
Let's take the derivative of the numerator.

128
00:05:47,685 --> 00:05:51,410
The derivative of negative 2 sine of x is negative

129
00:05:51,410 --> 00:05:54,100
2 cosine of x.

130
00:05:54,100 --> 00:05:57,850
And then, plus the derivative of 4 sine of 2x.

131
00:05:57,850 --> 00:06:01,520
Well, it's 2 times 4, which is 8.

132
00:06:01,519 --> 00:06:04,329
Times cosine of 2x.

133
00:06:04,329 --> 00:06:08,319
Derivative of sine of 2x is 2 cosine of 2x.

134
00:06:08,319 --> 00:06:10,269
And that first 2 gets multiplied by the

135
00:06:10,269 --> 00:06:11,839
4 to get the 8.

136
00:06:11,839 --> 00:06:17,029
And then the derivative of the denominator, derivative of sine

137
00:06:17,029 --> 00:06:19,139
of x is just cosine of x.

138
00:06:19,139 --> 00:06:21,879
So let's evaluate this character.

139
00:06:21,879 --> 00:06:24,620
So it looks like we've made some headway or maybe

140
00:06:24,620 --> 00:06:27,160
L'Hopital's rule stop applying here because we take the limit

141
00:06:27,160 --> 00:06:29,610
as x approaches 0 of cosine of x.

142
00:06:29,610 --> 00:06:30,650
That is 1.

143
00:06:30,649 --> 00:06:33,219
So we're definitely not going to get that indeterminate form,

144
00:06:33,220 --> 00:06:36,120
that 0/0 on this iteration.

145
00:06:36,120 --> 00:06:38,160
Let's see what happens to the numerator.

146
00:06:38,160 --> 00:06:41,950
We get negative 2 times cosine of 0.

147
00:06:41,949 --> 00:06:45,529
Well that's just negative 2 because cosine of 0 is 1.

148
00:06:45,529 --> 00:06:50,089
Plus 8 times cosine of 2x.

149
00:06:50,089 --> 00:06:53,589
Well, if x is 0, so it's going to be cosine of 0, which is 1.

150
00:06:53,589 --> 00:06:54,689
So it's just going to be an 8.

151
00:06:54,689 --> 00:06:56,990
So negative 2 plus 8.

152
00:06:56,990 --> 00:06:59,500
Well this thing right here, negative 2 plus 8 is 6.

153
00:06:59,500 --> 00:07:00,720
6 over 1.

154
00:07:00,720 --> 00:07:03,370
This whole thing is equal to 6.

155
00:07:03,370 --> 00:07:06,850
So L'Hopital's rule-- it applies to this last step.

156
00:07:06,850 --> 00:07:09,629
If this was the problem we were given and we said, hey, when we

157
00:07:09,629 --> 00:07:13,529
tried to apply the limit we get the limit as this numerator

158
00:07:13,529 --> 00:07:15,009
approaches 0 is 0.

159
00:07:15,009 --> 00:07:19,789
Limit as this denominator approaches 0 is 0.

160
00:07:19,790 --> 00:07:21,980
As the derivative of the numerator over the derivative

161
00:07:21,980 --> 00:07:25,600
of the denominator, that exists and it equals 6.

162
00:07:25,600 --> 00:07:28,590
So this limit must be equal to 6.

163
00:07:28,589 --> 00:07:31,739
Well if this limit is equal to 6, by the same argument, this

164
00:07:31,740 --> 00:07:33,340
limit is also going to be equal to 6.

165
00:07:33,339 --> 00:07:35,849
And by the same argument, this limit has got to

166
00:07:35,850 --> 00:07:40,090
also be equal to 6.

167
00:07:40,089 --> 00:07:41,769
And we're done.

168
00:07:41,769 --> 00:07:42,065