1
00:00:00,000 --> 00:00:00,670


2
00:00:00,670 --> 00:00:03,969
Now that we have hopefully a decent understanding of the

3
00:00:03,970 --> 00:00:09,279
squeeze theorem, we'll use that to prove that the limit-- I'll

4
00:00:09,279 --> 00:00:17,189
do it in yellow-- the limit as x approaches 0 of sine of

5
00:00:17,190 --> 00:00:22,280
x over x is equal to 1.

6
00:00:22,280 --> 00:00:25,089
And you must be bubbling over with anticipation now, because

7
00:00:25,089 --> 00:00:26,535
I've said this so many times.

8
00:00:26,535 --> 00:00:29,410
So let's do it, and actually, we have to go with-- obviously,

9
00:00:29,410 --> 00:00:32,219
they got our trigonometry-- and it's actually a visual proof.

10
00:00:32,219 --> 00:00:35,640
So let me draw at least the first and fourth quadrants

11
00:00:35,640 --> 00:00:37,439
of the unit circle.

12
00:00:37,439 --> 00:00:40,869
I'll do that in magenta.

13
00:00:40,869 --> 00:00:42,419
Let's see, let me see if I can-- I should

14
00:00:42,420 --> 00:00:44,219
draw it pretty big.

15
00:00:44,219 --> 00:00:44,839
Let me see.

16
00:00:44,840 --> 00:00:47,795
I should draw it like quite big.

17
00:00:47,795 --> 00:00:49,240
So I'll draw it like that.

18
00:00:49,240 --> 00:00:54,570


19
00:00:54,570 --> 00:00:55,780
That's close enough.

20
00:00:55,780 --> 00:00:59,030
And then let me draw the axis.

21
00:00:59,030 --> 00:01:02,920
So this is the x-axis, would look something like that.

22
00:01:02,920 --> 00:01:03,660
Sorry, that's the y-axis.

23
00:01:03,659 --> 00:01:06,179


24
00:01:06,180 --> 00:01:07,100
There you go.

25
00:01:07,099 --> 00:01:10,789
And then the x-axis, something like that.

26
00:01:10,790 --> 00:01:12,030
That's our unit circle.

27
00:01:12,030 --> 00:01:14,810


28
00:01:14,810 --> 00:01:16,219
There you go.

29
00:01:16,219 --> 00:01:18,689
Now let me draw a couple of other things.

30
00:01:18,689 --> 00:01:24,170
Let me draw a-- well, it is a radius, but I'm going to

31
00:01:24,170 --> 00:01:26,579
go beyond the unit circle.

32
00:01:26,579 --> 00:01:30,849


33
00:01:30,849 --> 00:01:34,969
So let's go like right there.

34
00:01:34,969 --> 00:01:39,689
Draw a couple of more things, just to set up this problem.

35
00:01:39,689 --> 00:01:41,649
Nope, that's not what I wanted to do.

36
00:01:41,650 --> 00:01:44,180
I wanted to do it right from this point.

37
00:01:44,180 --> 00:01:45,340
Right like that.

38
00:01:45,340 --> 00:01:49,260
And then right from this point, I want to do this.

39
00:01:49,260 --> 00:01:52,719


40
00:01:52,719 --> 00:01:55,560
And then I want to draw another right from that point.

41
00:01:55,560 --> 00:01:59,960
I'm going to do that.

42
00:01:59,959 --> 00:02:02,250
And now we are ready to go.

43
00:02:02,250 --> 00:02:03,640
So what did I say?

44
00:02:03,640 --> 00:02:05,859
This is the unit circle, right?

45
00:02:05,859 --> 00:02:07,569
So if that's the unit circle, what does it mean?

46
00:02:07,569 --> 00:02:09,780
It's a circle with a radius of 1.

47
00:02:09,780 --> 00:02:14,539
So the distance from here to here is 1.

48
00:02:14,539 --> 00:02:21,039
And now if this is an angle x in radiance, what's the length

49
00:02:21,039 --> 00:02:23,259
of this line right here?

50
00:02:23,259 --> 00:02:24,310
What's the length of that line?

51
00:02:24,310 --> 00:02:27,430
Well, by definition, sine of x is defined to be the

52
00:02:27,430 --> 00:02:30,830
y-coordinate of any point on the unit circle.

53
00:02:30,830 --> 00:02:32,070
So this is sine of x.

54
00:02:32,069 --> 00:02:35,269


55
00:02:35,270 --> 00:02:38,300
I'm going to run out of space, so let me draw an arrow.

56
00:02:38,300 --> 00:02:41,320
So this is-- that right there is sine of x.

57
00:02:41,319 --> 00:02:44,299


58
00:02:44,300 --> 00:02:46,270
Now let me ask you a slightly harder one.

59
00:02:46,270 --> 00:02:47,870
What's this length right here?

60
00:02:47,870 --> 00:02:51,370


61
00:02:51,370 --> 00:02:52,969
Well, let's think about it.

62
00:02:52,969 --> 00:02:53,879
What is tangent?

63
00:02:53,879 --> 00:02:56,370
Let's go back to our SOHCAHTOA definition of tangent.

64
00:02:56,370 --> 00:02:57,719
TOA.

65
00:02:57,719 --> 00:03:07,379
Tangent is equal to TOA: opposite over adjacent.

66
00:03:07,379 --> 00:03:08,909
So what is a tangent of x?

67
00:03:08,909 --> 00:03:14,460


68
00:03:14,460 --> 00:03:17,629
Well, it would be equal to-- we could take this-- if we say

69
00:03:17,629 --> 00:03:20,449
that this is the right triangle, it would be this

70
00:03:20,449 --> 00:03:26,109
length-- the opposite-- over the adjacent, right?

71
00:03:26,110 --> 00:03:28,440
So let's call this length over here, let's call

72
00:03:28,439 --> 00:03:30,379
this o for opposite.

73
00:03:30,379 --> 00:03:31,620
But what's the adjacent length?

74
00:03:31,620 --> 00:03:34,599
What's this base of this larger triangle?

75
00:03:34,599 --> 00:03:36,049
Well, it's the unit circle, right?

76
00:03:36,050 --> 00:03:40,660
So the distance from here to here-- that distance is

77
00:03:40,659 --> 00:03:41,829
also going to be 1, right?

78
00:03:41,830 --> 00:03:43,070
Because it's just a radius again.

79
00:03:43,069 --> 00:03:44,400
That's 1.

80
00:03:44,400 --> 00:03:48,980
So opposite over adjacent is equal to the tangent of x.

81
00:03:48,979 --> 00:03:52,349
But opposite over adjacent-- adjacent is just 1, right?

82
00:03:52,349 --> 00:03:56,280
So the opposite side, this side right here, it's going to be

83
00:03:56,280 --> 00:03:57,655
equal to the tangent of x.

84
00:03:57,655 --> 00:04:00,312
Or another way of saying it, tangent of x is equal to this

85
00:04:00,312 --> 00:04:02,930
side over 1, or tangent of x is equal to this side.

86
00:04:02,930 --> 00:04:04,700
So let me write that down.

87
00:04:04,699 --> 00:04:08,079
That side is equal to the tangent of x.

88
00:04:08,080 --> 00:04:11,840


89
00:04:11,840 --> 00:04:17,769
Now, let's think about the area of a couple of parts of this

90
00:04:17,769 --> 00:04:18,729
figure that I've drawn here.

91
00:04:18,730 --> 00:04:20,580
Maybe I should have drawn it a little bigger, but I think

92
00:04:20,579 --> 00:04:22,560
we'll be able to do it.

93
00:04:22,560 --> 00:04:25,209
So first let me pick a relatively small triangle.

94
00:04:25,209 --> 00:04:27,579
So let's do this triangle right here.

95
00:04:27,579 --> 00:04:29,870
I'll trace it in green.

96
00:04:29,870 --> 00:04:34,250
So this triangle that I'm tracing in green-- what is

97
00:04:34,250 --> 00:04:37,730
the area of that triangle?

98
00:04:37,730 --> 00:04:40,910
Well, that's going to be 1/2 times base times height.

99
00:04:40,910 --> 00:04:49,850
So it's 1/2 times the base, which is 1.

100
00:04:49,850 --> 00:04:50,110
Right?

101
00:04:50,110 --> 00:04:51,600
It's this whole triangle.

102
00:04:51,600 --> 00:04:53,240
And then what's the height of it?

103
00:04:53,240 --> 00:04:55,740
Well, we just figured out that this height right here, that

104
00:04:55,740 --> 00:04:57,689
this height is sine of x.

105
00:04:57,689 --> 00:04:58,779
Times sine of x.

106
00:04:58,779 --> 00:05:01,879


107
00:05:01,879 --> 00:05:04,649
So that's this green triangle here, right?

108
00:05:04,649 --> 00:05:09,039
Now, what is the area of-- not that green triangle.

109
00:05:09,040 --> 00:05:11,180
Let me do it in another color.

110
00:05:11,180 --> 00:05:16,480
Let me do it in-- oh, I'll do it in red.

111
00:05:16,480 --> 00:05:20,540
What is the area of this pi?

112
00:05:20,540 --> 00:05:23,470
This pi right here.

113
00:05:23,470 --> 00:05:24,270
That pi.

114
00:05:24,269 --> 00:05:26,620
Hope you see-- well, that's not a different enough color.

115
00:05:26,620 --> 00:05:29,439
So, this pi right here.

116
00:05:29,439 --> 00:05:30,329
Or I'm going there.

117
00:05:30,329 --> 00:05:32,050
And then I'm going on the arc.

118
00:05:32,050 --> 00:05:34,629
So it's a little bit bigger than the triangle we

119
00:05:34,629 --> 00:05:35,639
just figured out, right?

120
00:05:35,639 --> 00:05:37,129
It's always going to be a little bit bigger, because it

121
00:05:37,129 --> 00:05:41,219
includes this area between that triangle and the arc, right?

122
00:05:41,220 --> 00:05:42,890
What is the area of that arc?

123
00:05:42,889 --> 00:05:45,599


124
00:05:45,600 --> 00:05:54,500
Well, if this angle is x-- it's x radiance-- what fraction

125
00:05:54,500 --> 00:05:56,660
of that is out of the entire unit circle?

126
00:05:56,660 --> 00:05:59,920
Well, there are 2 pi radians in a total unit circle, right?

127
00:05:59,920 --> 00:06:04,220
So this area right here is going to be equal to what?

128
00:06:04,220 --> 00:06:09,690
It's going to be equal to the fraction x is of the total

129
00:06:09,689 --> 00:06:14,269
radians in the unit circle, right?

130
00:06:14,269 --> 00:06:17,060
So it's x radians over 2 pi radians in the

131
00:06:17,060 --> 00:06:18,449
entire unit circle.

132
00:06:18,449 --> 00:06:21,134
So that's kind of the fraction that this is of-- you know, if

133
00:06:21,134 --> 00:06:23,769
you did it in degrees-- the fraction that this is over 360

134
00:06:23,769 --> 00:06:26,349
degrees, times the area of the whole circle, right?

135
00:06:26,350 --> 00:06:28,490
This tells us what fraction we are of the circle, and we're

136
00:06:28,490 --> 00:06:30,280
going to want to multiply that times the area of

137
00:06:30,279 --> 00:06:31,349
the whole circle.

138
00:06:31,350 --> 00:06:33,450
Well, what's the area of the whole circle?

139
00:06:33,449 --> 00:06:38,659
Well, area is pi r squared, the radius is 1, right?

140
00:06:38,660 --> 00:06:42,100
So the area of the entire circle is just pi.

141
00:06:42,100 --> 00:06:45,250


142
00:06:45,250 --> 00:06:47,699
Pi r squared, r is 1, so the area of the circle-- so the

143
00:06:47,699 --> 00:06:51,959
area of this wedge right here, is just going to be equal to--

144
00:06:51,959 --> 00:06:55,209
these pi's cancel out-- it's equal to x over 2.

145
00:06:55,209 --> 00:07:00,859
So that first small triangle, that green triangle

146
00:07:00,860 --> 00:07:04,050
we did, is sine of x.

147
00:07:04,050 --> 00:07:07,629
1/2 sine of x, that's the area of that green triangle.

148
00:07:07,629 --> 00:07:11,389
Then the slightly larger area of this wedge is-- we figured

149
00:07:11,389 --> 00:07:12,639
out just now-- is x over 2.

150
00:07:12,639 --> 00:07:15,629
And now let's take the area of that larger triangle,

151
00:07:15,629 --> 00:07:17,019
of this big triangle here.

152
00:07:17,019 --> 00:07:18,859
And that may be the most obvious.

153
00:07:18,860 --> 00:07:20,129
So 1/2 base times height.

154
00:07:20,129 --> 00:07:24,949
So that's 1/2-- the base is 1 again-- 1 times the

155
00:07:24,949 --> 00:07:28,719
height, is tangent of x.

156
00:07:28,720 --> 00:07:33,900
Equal to 1/2 tangent of x.

157
00:07:33,899 --> 00:07:36,639
Now, it should be clear just looking at this diagram, no

158
00:07:36,639 --> 00:07:40,349
matter where I drew this top line, that this green triangle

159
00:07:40,350 --> 00:07:43,689
has a smaller area than this wedge, which has a smaller area

160
00:07:43,689 --> 00:07:45,329
than this large triangle.

161
00:07:45,329 --> 00:07:45,959
Right?

162
00:07:45,959 --> 00:07:48,509
So let's write an inequality that says that.

163
00:07:48,509 --> 00:07:54,829
The green triangle-- the area of the green triangle-- so 1/2

164
00:07:54,829 --> 00:07:58,899
the sine of x, that's the area of the green triangle-- it's

165
00:07:58,899 --> 00:08:04,029
less than the area of this wedge.

166
00:08:04,029 --> 00:08:06,359
So that's x over 2.

167
00:08:06,360 --> 00:08:09,540
And they're both less than the area of this large

168
00:08:09,540 --> 00:08:11,200
triangle, right?

169
00:08:11,199 --> 00:08:13,860
Which is 1/2 tangent of x.

170
00:08:13,860 --> 00:08:16,920


171
00:08:16,920 --> 00:08:18,009
Now when is this true?

172
00:08:18,009 --> 00:08:21,159
This is true as long as we're in the first quadrant, right?

173
00:08:21,160 --> 00:08:23,400
As long as we're in the first quadrant.

174
00:08:23,399 --> 00:08:25,959
It's also almost true if we go into the fourth quadrant,

175
00:08:25,959 --> 00:08:28,310
except then the sine of x becomes negative, the tangent

176
00:08:28,310 --> 00:08:31,129
of x becomes negative, and x becomes negative.

177
00:08:31,129 --> 00:08:33,299
But if we take the absolute value of everything, it still

178
00:08:33,299 --> 00:08:34,899
holds in the fourth quadrant.

179
00:08:34,899 --> 00:08:37,449
Because if you go negative, as long as we take the absolute

180
00:08:37,450 --> 00:08:39,900
value, then the distance will still hold and we still have

181
00:08:39,899 --> 00:08:42,120
positive areas and all that kind of thing.

182
00:08:42,120 --> 00:08:47,060
So since my goal is to take the limit as x approaches 0, and I

183
00:08:47,059 --> 00:08:49,539
want to take the limit-- in order for this limit to be

184
00:08:49,539 --> 00:08:53,259
defined in general, it has to be true from both the positive

185
00:08:53,259 --> 00:08:54,319
and the negative side.

186
00:08:54,320 --> 00:08:56,170
Let's take the absolute value of both sides of this.

187
00:08:56,169 --> 00:08:57,750
And hopefully this makes sense to you.

188
00:08:57,750 --> 00:08:59,950
If I were to draw the line down here-- and this would be the

189
00:08:59,950 --> 00:09:02,720
sine of x, and that would be the tangent of x-- as long as

190
00:09:02,720 --> 00:09:04,350
you took the absolute value of everything, you're essentially

191
00:09:04,350 --> 00:09:06,370
just doing the same thing as in the first quadrant.

192
00:09:06,370 --> 00:09:10,149
So let's take the absolute value of everything.

193
00:09:10,149 --> 00:09:14,019
And that shouldn't change anything, especially if you're

194
00:09:14,019 --> 00:09:14,750
in the first quadrant.

195
00:09:14,750 --> 00:09:16,190
And you might want to think about it a little bit, why

196
00:09:16,190 --> 00:09:18,290
it doesn't change anything in the second quadrant.

197
00:09:18,289 --> 00:09:19,509
So we have this inequality.

198
00:09:19,509 --> 00:09:21,269
Let's see if we can play around with this.

199
00:09:21,269 --> 00:09:22,939
So first of all, let's just multiply everything by 2

200
00:09:22,940 --> 00:09:25,180
and get rid of the 1/2's.

201
00:09:25,179 --> 00:09:33,109
So we get absolute value of sine of x is less than absolute

202
00:09:33,110 --> 00:09:36,769
value of x, which is less than the absolute value of

203
00:09:36,769 --> 00:09:38,500
the tangent of x.

204
00:09:38,500 --> 00:09:41,419
I hope I didn't confuse you by taking the absolute value.

205
00:09:41,419 --> 00:09:44,939
That original inequality I wrote was completely valid in

206
00:09:44,940 --> 00:09:48,310
the first quadrant, but since I want this inequality to be true

207
00:09:48,309 --> 00:09:50,509
in the first and fourth quadrants, because I'm taking

208
00:09:50,509 --> 00:09:53,259
the limit as x approaches 0 from both sides, I put that

209
00:09:53,259 --> 00:09:54,120
absolute value there.

210
00:09:54,120 --> 00:09:56,539
So you could draw the line down there and do everything we did

211
00:09:56,539 --> 00:09:58,469
up there in the fourth quadrant, but just take

212
00:09:58,470 --> 00:10:01,170
the absolute value and it should work out the same.

213
00:10:01,169 --> 00:10:02,389
Anyway, back to the problem.

214
00:10:02,389 --> 00:10:03,730
So we have this inequality.

215
00:10:03,730 --> 00:10:06,629
And I'm running out of space, so let me erase some

216
00:10:06,629 --> 00:10:07,710
of this stuff up here.

217
00:10:07,710 --> 00:10:11,280


218
00:10:11,279 --> 00:10:12,569
Erase.

219
00:10:12,570 --> 00:10:12,760
Erase.

220
00:10:12,759 --> 00:10:15,419


221
00:10:15,419 --> 00:10:17,939
Nope, that doesn't erase.

222
00:10:17,940 --> 00:10:18,960
OK.

223
00:10:18,960 --> 00:10:21,400
That should erase.

224
00:10:21,399 --> 00:10:23,230
OK.

225
00:10:23,230 --> 00:10:25,980
So we could erase everything that took us so far.

226
00:10:25,980 --> 00:10:28,122
We can't forget this though.

227
00:10:28,121 --> 00:10:30,309
This gives a lot of space.

228
00:10:30,309 --> 00:10:32,429
OK.

229
00:10:32,429 --> 00:10:35,699
So let's take this, and let's take that expression, and

230
00:10:35,700 --> 00:10:38,460
divide all of the sides.

231
00:10:38,460 --> 00:10:40,190
You know, and it has three sides, a left,

232
00:10:40,190 --> 00:10:41,100
middle, and right.

233
00:10:41,100 --> 00:10:43,730
Let's divide them all by the absolute value of sine of x.

234
00:10:43,730 --> 00:10:45,960
And since we know that the absolute value of sine of x is

235
00:10:45,960 --> 00:10:49,639
a positive number, we know that these less than signs

236
00:10:49,639 --> 00:10:51,154
don't change, right?

237
00:10:51,154 --> 00:10:52,259
So let's do that.

238
00:10:52,259 --> 00:10:55,210
So the absolute value of the sine of x divided by the

239
00:10:55,210 --> 00:10:57,585
absolute value of the sine of x, well, that equals 1.

240
00:10:57,585 --> 00:11:00,440


241
00:11:00,440 --> 00:11:04,230
Which is less than the absolute value of x divided by the

242
00:11:04,230 --> 00:11:05,870
absolute value of sine of x.

243
00:11:05,870 --> 00:11:08,649


244
00:11:08,649 --> 00:11:12,419
Which is less than-- what's the absolute value of tan-- so, all

245
00:11:12,419 --> 00:11:15,849
I'm doing is I'm taking the absolute value of sine of x,

246
00:11:15,850 --> 00:11:20,909
absolute value of sine of x, absolute value of sine of x.

247
00:11:20,909 --> 00:11:23,350
So what's the absolute value of the tangent of x divided by the

248
00:11:23,350 --> 00:11:26,580
absolute value of the sine of x?

249
00:11:26,580 --> 00:11:32,930
Well, tangent is just sine over cosine.

250
00:11:32,929 --> 00:11:35,579
So that's equal to-- so, just do this part right here.

251
00:11:35,580 --> 00:11:40,430
That's equal to sine over cosine divided by sine.

252
00:11:40,429 --> 00:11:41,894
And you know, you could say that that's the same thing

253
00:11:41,894 --> 00:11:42,679
as the absolute value.

254
00:11:42,679 --> 00:11:45,359
And the absolute value divided by the absolute value.

255
00:11:45,360 --> 00:11:46,899
So what are you left with?

256
00:11:46,899 --> 00:11:50,620
Well, you're just left with 1 over-- this cancels out with

257
00:11:50,620 --> 00:11:54,549
this, that becomes a 1-- 1 over the absolute value

258
00:11:54,549 --> 00:11:57,719
of the cosine of x.

259
00:11:57,720 --> 00:12:00,629
So you might feel we're getting close.

260
00:12:00,629 --> 00:12:03,700
Because this looks a lot like this, it's just inverted.

261
00:12:03,700 --> 00:12:06,050
So to get to this, let's invert it.

262
00:12:06,049 --> 00:12:07,750
And to invert it, what happens?

263
00:12:07,750 --> 00:12:10,090
Well, first of all, what happens when you invert 1?

264
00:12:10,090 --> 00:12:13,879
Well, 1/1 is just 1.

265
00:12:13,879 --> 00:12:17,759
But when you invert both sides of an inequality, you switch

266
00:12:17,759 --> 00:12:18,960
the inequality, right?

267
00:12:18,960 --> 00:12:22,480
And if that doesn't make sense to you, think about this.

268
00:12:22,480 --> 00:12:26,870
You know, if I say 1/2 is less than 2, and I invert both sides

269
00:12:26,870 --> 00:12:32,019
of that, I get 2 is greater than 1/2.

270
00:12:32,019 --> 00:12:33,829
So that hopefully gives you a little intuition.

271
00:12:33,830 --> 00:12:37,379
So if I'm inverting all of the sides of this inequality, I

272
00:12:37,379 --> 00:12:40,210
have to switch the inequality signs.

273
00:12:40,210 --> 00:12:48,129
So 1 is greater than absolute value of sine of x, over the

274
00:12:48,129 --> 00:12:52,414
absolute value of x, which is greater than absolute

275
00:12:52,414 --> 00:12:55,399
value of cosine of x.

276
00:12:55,399 --> 00:12:58,429
Now let me ask you a question.

277
00:12:58,429 --> 00:13:01,329
The absolute value of sine of x over-- well, first

278
00:13:01,330 --> 00:13:04,030
of all, sine of x over x.

279
00:13:04,029 --> 00:13:10,139
Will there ever be a time when sine of x over x is-- in the

280
00:13:10,139 --> 00:13:12,730
first or the fourth quadrant-- is there ever a time that

281
00:13:12,730 --> 00:13:17,110
sine of x over x is a negative expression?

282
00:13:17,110 --> 00:13:19,680
Well, in the first quadrant, sine of x is positive,

283
00:13:19,679 --> 00:13:20,909
and x is positive.

284
00:13:20,909 --> 00:13:22,959
So a positive divided by a positive is

285
00:13:22,960 --> 00:13:24,090
going to be positive.

286
00:13:24,090 --> 00:13:29,610
And in the fourth quadrant, sine of x is negative, y is

287
00:13:29,610 --> 00:13:32,389
negative, and the angle is negative, so x is

288
00:13:32,389 --> 00:13:33,399
also negative.

289
00:13:33,399 --> 00:13:37,240
So in the fourth quadrant, sine of x over x is going to be a

290
00:13:37,240 --> 00:13:39,009
negative divided by a negative.

291
00:13:39,009 --> 00:13:40,850
So it's going to be a positive again.

292
00:13:40,850 --> 00:13:45,070
So sine of x over x is always going to be a positive.

293
00:13:45,070 --> 00:13:48,120
So the absolute value signs are kind of redundant.

294
00:13:48,120 --> 00:13:53,990
So we could write 1 is greater than sine of x over x.

295
00:13:53,990 --> 00:13:57,250
And the same logic, in the first and fourth quadrants--

296
00:13:57,250 --> 00:13:58,539
and that's where we're dealing with.

297
00:13:58,539 --> 00:14:05,860
We're dealing with minus pi over 2 is less than x, which

298
00:14:05,860 --> 00:14:08,000
is less than pi over 2.

299
00:14:08,000 --> 00:14:09,960
So we're going from minus pi over 2 all

300
00:14:09,960 --> 00:14:11,310
the way to pi over 2.

301
00:14:11,309 --> 00:14:13,509
So we're in the fourth and first quadrant.

302
00:14:13,509 --> 00:14:15,049
Is cosine of x ever negative?

303
00:14:15,049 --> 00:14:17,829
Well, cosine is the x value, and the x-- by definition, in

304
00:14:17,830 --> 00:14:19,653
the first and fourth quadrants-- the x value

305
00:14:19,653 --> 00:14:20,879
is always positive.

306
00:14:20,879 --> 00:14:23,480
So if this is always positive, we can get rid of the

307
00:14:23,480 --> 00:14:27,940
absolute value signs there, and just write that.

308
00:14:27,940 --> 00:14:33,430
And now, we are ready to use the squeeze theorem.

309
00:14:33,429 --> 00:14:36,059
Let me erase all of this down here now.

310
00:14:36,059 --> 00:14:39,579


311
00:14:39,580 --> 00:14:42,500
So let me ask you a question.

312
00:14:42,500 --> 00:14:50,169
What is the limit, as x approaches 0, of

313
00:14:50,169 --> 00:14:52,750
the function 1?

314
00:14:52,750 --> 00:14:55,509
Well, the function 1 is always equal to 1.

315
00:14:55,509 --> 00:14:57,460
So I can set the limit as x approaches infinity, the limit

316
00:14:57,460 --> 00:14:59,310
as x approaches pi, anything.

317
00:14:59,309 --> 00:15:02,750
This is always going to be equal to 1.

318
00:15:02,750 --> 00:15:05,620
So as x approaches 0, this is equal to 1.

319
00:15:05,620 --> 00:15:14,570
And then what is the limit, as x approaches 0, of cosine of x?

320
00:15:14,570 --> 00:15:15,620
Well, that's easy, too.

321
00:15:15,620 --> 00:15:18,169
As x approaches 0, cosine of 0 is just 1-- and as you get,

322
00:15:18,169 --> 00:15:22,620
you know, it's a continuous function-- so the limit is 1.

323
00:15:22,620 --> 00:15:25,629
So we are ready to use the squeeze theorem.

324
00:15:25,629 --> 00:15:31,990
As we approach 0, as x approaches 0, this

325
00:15:31,990 --> 00:15:33,769
function approaches 1.

326
00:15:33,769 --> 00:15:35,649
This function approaches 1.

327
00:15:35,649 --> 00:15:38,350
And this function, this expression, is in

328
00:15:38,350 --> 00:15:39,870
between the two.

329
00:15:39,870 --> 00:15:45,299
And if it's in between the two, as we approach-- this is

330
00:15:45,299 --> 00:15:48,289
approaching 1 as we approach 0, this is approaching 1 as we

331
00:15:48,289 --> 00:15:51,469
approach 0, and this is in between them, so it also has to

332
00:15:51,470 --> 00:15:53,790
approach 1 as we approach 0.

333
00:15:53,789 --> 00:15:58,339
And so we are using the squeeze theorem based on this and this.

334
00:15:58,340 --> 00:16:00,090
And you could say, you know, therefore by the squeeze

335
00:16:00,090 --> 00:16:03,740
theorem, because this is true, this is true, and this is true,

336
00:16:03,740 --> 00:16:11,009
sine of x over x, the limit as x approaches 0, is equal to 1.

337
00:16:11,009 --> 00:16:13,309
So hopefully that gave you the intuition.

338
00:16:13,309 --> 00:16:18,909
That another way to view it, as this line gets smaller and

339
00:16:18,909 --> 00:16:24,409
smaller as it approaches 0, as x approaches zero, that this

340
00:16:24,409 --> 00:16:27,299
area and this area converge, so the area in between kind of has

341
00:16:27,299 --> 00:16:30,659
to converge to the both of them.

342
00:16:30,659 --> 00:16:32,699
And if you want to see it graphically, I've

343
00:16:32,700 --> 00:16:34,490
graphed it here.

344
00:16:34,490 --> 00:16:37,284
Let me see if I can graph this thing.

345
00:16:37,284 --> 00:16:40,610
I'll show you the graph.

346
00:16:40,610 --> 00:16:42,850
Just so you believe me.

347
00:16:42,850 --> 00:16:47,269
So we said that 1 is always greater than sine of x, which

348
00:16:47,269 --> 00:16:52,629
is always greater than cosine of x, between negative pi

349
00:16:52,629 --> 00:16:53,439
over 2 and pi over 2.

350
00:16:53,440 --> 00:16:56,740
And of course, this isn't defined at x is equal to 0.

351
00:16:56,740 --> 00:16:58,350
But we can figure out the limit.

352
00:16:58,350 --> 00:16:59,529
So there we have it.

353
00:16:59,529 --> 00:17:03,129
This blue line right here, that's the function 1.

354
00:17:03,129 --> 00:17:06,650
That's y is equal to 1.

355
00:17:06,650 --> 00:17:09,440
This light blue line right here is cosine of x.

356
00:17:09,440 --> 00:17:11,430
And this is the graph of sine of x over x.

357
00:17:11,430 --> 00:17:15,480
And you can see that I actually typed it in.

358
00:17:15,480 --> 00:17:22,059
So sine of x over x, between negative pi over 2 and pi over

359
00:17:22,059 --> 00:17:27,169
2, or the fourth and the first quadrants, the red line

360
00:17:27,170 --> 00:17:29,100
is always in between.

361
00:17:29,099 --> 00:17:32,750
It's always in between the dark blue and the light blue line.

362
00:17:32,750 --> 00:17:34,970
And so this is just an intuition of what happens

363
00:17:34,970 --> 00:17:36,180
with the squeeze theorem.

364
00:17:36,180 --> 00:17:39,390
We know that the limit, as this light blue line

365
00:17:39,390 --> 00:17:40,860
approaches 0, is 1.

366
00:17:40,859 --> 00:17:43,299
And we know the limit as this top dark blue line

367
00:17:43,299 --> 00:17:45,500
approaches 0 is 1.

368
00:17:45,500 --> 00:17:47,779
And this red line is always in between it, so it

369
00:17:47,779 --> 00:17:49,240
also approaches 1.

370
00:17:49,240 --> 00:17:50,599
So there you have it.

371
00:17:50,599 --> 00:17:53,369
The proof, using the squeeze theorem, and a little bit of

372
00:17:53,369 --> 00:17:57,959
visual trigonometry, of why the limit, as x approaches 0, of

373
00:17:57,960 --> 00:18:00,970
sine of x over x is equal to 1.

374
00:18:00,970 --> 00:18:03,900
I hope I haven't confused you.

375
00:18:03,900 --> 00:18:04,400