1
00:00:00,000 --> 00:00:00,750


2
00:00:00,750 --> 00:00:02,790
I've been asked to do some more limit problems.

3
00:00:02,790 --> 00:00:04,429
So let's do some more.

4
00:00:04,429 --> 00:00:04,859
So let's see.

5
00:00:04,860 --> 00:00:06,150
What's the first problem I have?

6
00:00:06,150 --> 00:00:19,329
It says the limit as x approaches 0 of x minus 2 times

7
00:00:19,329 --> 00:00:26,969
the absolute value of x over the absolute value of x.

8
00:00:26,969 --> 00:00:29,729
Now what immediately confuses most people is this

9
00:00:29,730 --> 00:00:30,780
absolute value thing.

10
00:00:30,780 --> 00:00:32,960
Because how do you deal with it?

11
00:00:32,960 --> 00:00:34,490
Because you can't just subtract.

12
00:00:34,490 --> 00:00:36,410
If this was an x, it would be a simple problem.

13
00:00:36,409 --> 00:00:37,559
Because then you could just simplify it.

14
00:00:37,560 --> 00:00:40,490
You could subtract this 2x from this x and then divide by x.

15
00:00:40,490 --> 00:00:41,550
And you'd get an answer.

16
00:00:41,549 --> 00:00:43,739
But this has an absolute value, so you can't just

17
00:00:43,740 --> 00:00:45,539
directly manipulate it.

18
00:00:45,539 --> 00:00:48,629
Well what if we're able to get rid of this absolute value?

19
00:00:48,630 --> 00:00:49,620
How do we do that?

20
00:00:49,619 --> 00:00:53,339
Well, in order for this limit to exist-- in order for

21
00:00:53,340 --> 00:00:54,250
this limit-- I don't know.

22
00:00:54,250 --> 00:00:56,869
Let's say it equals l, right?

23
00:00:56,869 --> 00:00:58,820
And l is what we have to solve for.

24
00:00:58,820 --> 00:01:02,390
In order for that limit to exist, then the limit to x

25
00:01:02,390 --> 00:01:05,230
approaches 0 also has to exist from the positive

26
00:01:05,230 --> 00:01:06,180
and the negative sides.

27
00:01:06,180 --> 00:01:07,040
What do I mean?

28
00:01:07,040 --> 00:01:10,380
Well that means that this must also be true.

29
00:01:10,379 --> 00:01:15,569
The limit as x approaches 0 from the right-hand side, from

30
00:01:15,569 --> 00:01:23,299
positive numbers, of x minus 2 times the absolute value of x

31
00:01:23,299 --> 00:01:27,209
over the absolute value of x has to be equal to l.

32
00:01:27,209 --> 00:01:33,239
And then we also know that the limit as x approaches 0 from

33
00:01:33,239 --> 00:01:38,060
the negative side of the same function, 2 times absolute

34
00:01:38,060 --> 00:01:42,680
value of x over the absolute value of x, is also going

35
00:01:42,680 --> 00:01:43,830
to be equal to l, right?

36
00:01:43,829 --> 00:01:47,359
If this-- If we-- As we approach 0 from the left,

37
00:01:47,359 --> 00:01:48,254
we get some number.

38
00:01:48,254 --> 00:01:51,789
And as we approach 0 from the right, we get the same number,

39
00:01:51,790 --> 00:01:54,540
then we know that no matter what direction we approach 0

40
00:01:54,540 --> 00:01:58,140
from, which is this situation, we approach that number.

41
00:01:58,140 --> 00:01:59,385
So let's figure out these two limits.

42
00:01:59,385 --> 00:02:01,250
And if they have the same answer, then we have the answer

43
00:02:01,250 --> 00:02:03,099
to our original problem.

44
00:02:03,099 --> 00:02:05,989
So how can we simplify this as x approaches 0 from

45
00:02:05,989 --> 00:02:08,180
the positive side?

46
00:02:08,180 --> 00:02:13,590
Well that's the same thing as the limit as x approaches

47
00:02:13,590 --> 00:02:14,840
0 from the positive side.

48
00:02:14,840 --> 00:02:17,750
Well if we know we're approaching 0 from the positive

49
00:02:17,750 --> 00:02:19,210
side, what do we know about x?

50
00:02:19,210 --> 00:02:21,500
We know that x is positive, right?

51
00:02:21,500 --> 00:02:23,530
It's approaching from the right.

52
00:02:23,530 --> 00:02:25,439
So if x is positive, we can get rid of these

53
00:02:25,439 --> 00:02:26,449
absolute value signs.

54
00:02:26,449 --> 00:02:32,469
So we just get x minus 2x over x.

55
00:02:32,469 --> 00:02:38,069
And so that equals the limit as x approaches 0 from

56
00:02:38,069 --> 00:02:39,269
the positive side.

57
00:02:39,270 --> 00:02:39,590
Let's see.

58
00:02:39,590 --> 00:02:42,759
That's x minus 2x.

59
00:02:42,759 --> 00:02:46,310
This is minus x, right?

60
00:02:46,310 --> 00:02:47,150
x minus x.

61
00:02:47,150 --> 00:02:53,430
And then minus x divided by x is minus 1.

62
00:02:53,430 --> 00:02:55,590
So the limit as x approaches 0 from the positive

63
00:02:55,590 --> 00:02:56,420
side of minus 1.

64
00:02:56,419 --> 00:02:58,119
Well that's just going to equal minus 1.

65
00:02:58,120 --> 00:03:05,120


66
00:03:05,120 --> 00:03:05,480
OK.

67
00:03:05,479 --> 00:03:07,969
Now let's take this case down here when we're approaching

68
00:03:07,969 --> 00:03:09,699
from the negative side.

69
00:03:09,699 --> 00:03:12,829


70
00:03:12,830 --> 00:03:15,400
So how can we think about this?

71
00:03:15,400 --> 00:03:22,240
What is the absolute value of x if x is a negative number?

72
00:03:22,240 --> 00:03:25,930
Well the absolute value of x, if x is a negative number,

73
00:03:25,930 --> 00:03:29,439
is going to be the negative of x, right?

74
00:03:29,439 --> 00:03:33,490
If x-- Think of it this way: if x is negative 1, when we take

75
00:03:33,490 --> 00:03:35,680
the absolute value of x, we're essentially just multiplying

76
00:03:35,680 --> 00:03:37,610
x times negative 1 again.

77
00:03:37,610 --> 00:03:45,160
So another way to rewrite this is: this is equal to the limit

78
00:03:45,159 --> 00:03:52,060
as x approaches 0 from the negative side of

79
00:03:52,060 --> 00:03:58,670
x minus 2 times what?

80
00:03:58,669 --> 00:04:01,649
The absolute value of x is the same thing as negative x.

81
00:04:01,650 --> 00:04:04,539


82
00:04:04,539 --> 00:04:06,000
Hopefully that makes sense to you, right?

83
00:04:06,000 --> 00:04:08,069
If we're dealing with negative numbers, then taking the

84
00:04:08,069 --> 00:04:11,159
absolute value of a negative number is the same thing as

85
00:04:11,159 --> 00:04:13,710
multiplying that negative number times the negative to

86
00:04:13,710 --> 00:04:17,689
essentially make it into a positive number, right?

87
00:04:17,689 --> 00:04:19,625
And then, of course, the absolute value of x in the

88
00:04:19,625 --> 00:04:21,449
denominator, since we're dealing with a negative number,

89
00:04:21,449 --> 00:04:23,259
is also equal to negative x.

90
00:04:23,259 --> 00:04:26,420


91
00:04:26,420 --> 00:04:28,040
And let's see if we can simplify that.

92
00:04:28,040 --> 00:04:33,800
So that means that the limit as x approaches 0 from the

93
00:04:33,800 --> 00:04:36,379
negative side-- Let's see.

94
00:04:36,379 --> 00:04:39,600
I get x minus 2 times negative x.

95
00:04:39,600 --> 00:04:41,230
So these negatives cancel out.

96
00:04:41,230 --> 00:04:45,379
So I get x plus 2x, right?

97
00:04:45,379 --> 00:04:48,495
So I get 3x over minus x.

98
00:04:48,495 --> 00:04:56,680


99
00:04:56,680 --> 00:05:00,449
That equals minus 3.

100
00:05:00,449 --> 00:05:02,099
So this is interesting.

101
00:05:02,100 --> 00:05:05,200
I am approaching a different number when I approach it

102
00:05:05,199 --> 00:05:06,949
from the left-hand side.

103
00:05:06,949 --> 00:05:10,500
When I approached this function from the left-hand side or

104
00:05:10,500 --> 00:05:11,850
from the right-hand side.

105
00:05:11,850 --> 00:05:15,160
So it looks like this limit doesn't exist.

106
00:05:15,160 --> 00:05:15,880
Let's confirm that.

107
00:05:15,879 --> 00:05:22,680
Let me actually-- Let me see if I can-- Let me get the graphing

108
00:05:22,680 --> 00:05:25,250
calculator and confirm.

109
00:05:25,250 --> 00:05:26,639
Let me type it in.

110
00:05:26,639 --> 00:05:32,635
So x minus-- I'll show you what I'm doing --x-- so you

111
00:05:32,636 --> 00:05:38,110
don't get bored --minus absolute value of x.

112
00:05:38,110 --> 00:05:43,080
And what am I-- Divided by the absolute value of x.

113
00:05:43,079 --> 00:05:48,240


114
00:05:48,240 --> 00:05:49,129
Let's see.

115
00:05:49,129 --> 00:05:51,769
Graph.

116
00:05:51,769 --> 00:05:53,569
And what does it have here?

117
00:05:53,569 --> 00:05:54,170
Zoom out.

118
00:05:54,170 --> 00:06:01,030


119
00:06:01,029 --> 00:06:03,529
So this is-- Is this right?

120
00:06:03,529 --> 00:06:08,250
x minus the absolute value of x-- Oh, sorry.

121
00:06:08,250 --> 00:06:10,389
It's x minus 2 times the absolute value of x.

122
00:06:10,389 --> 00:06:15,050


123
00:06:15,050 --> 00:06:16,170
Graph.

124
00:06:16,170 --> 00:06:18,189
There you go.

125
00:06:18,189 --> 00:06:19,100
Even the-- oh.

126
00:06:19,100 --> 00:06:20,689
I don't think you can see it yet.

127
00:06:20,689 --> 00:06:24,060
--the graphing calculator confirms the work we did.

128
00:06:24,060 --> 00:06:25,899
Although it connects them and it makes you think the it

129
00:06:25,899 --> 00:06:26,709
somehow approaches here.

130
00:06:26,709 --> 00:06:28,489
But that's just because it picks points and

131
00:06:28,490 --> 00:06:29,879
just plots them.

132
00:06:29,879 --> 00:06:32,219
So as you see, as you approach from the right-hand side, you

133
00:06:32,220 --> 00:06:36,020
approach negative 1, right?

134
00:06:36,019 --> 00:06:37,789
Well actually you're at negative 1 the whole time

135
00:06:37,790 --> 00:06:38,770
on the right-hand side.

136
00:06:38,769 --> 00:06:40,209
And as you approach from the left-hand side, you

137
00:06:40,209 --> 00:06:41,409
approach negative 3.

138
00:06:41,410 --> 00:06:44,090
So the limit does not exist at x is equal 0.

139
00:06:44,089 --> 00:06:47,289
You would say l is undefined.

140
00:06:47,290 --> 00:06:50,700
You know, there's a little dirty secret about limits.

141
00:06:50,699 --> 00:06:53,269
It's not a dirty secret, but in theory you should never

142
00:06:53,269 --> 00:06:54,259
get a limit problem wrong.

143
00:06:54,259 --> 00:06:55,170
And why?

144
00:06:55,170 --> 00:06:57,509
Well you should be able to solve it analytically.

145
00:06:57,509 --> 00:06:59,860
But if you don't know how to solve it analytically, just

146
00:06:59,860 --> 00:07:02,080
put it really really small numbers here.

147
00:07:02,079 --> 00:07:06,810
Try out-- If you put in-- Try 0.0001.

148
00:07:06,810 --> 00:07:10,870
Try numbers that are just slightly larger then whatever

149
00:07:10,870 --> 00:07:12,209
your limit number is.

150
00:07:12,209 --> 00:07:13,409
And then slightly smaller.

151
00:07:13,410 --> 00:07:15,630
And then, just numerically, if you have a calculator, see

152
00:07:15,629 --> 00:07:16,370
what it's approaching.

153
00:07:16,370 --> 00:07:18,370
And sometimes you don't even need a calculator.

154
00:07:18,370 --> 00:07:20,740
You could probably calculate this in your head with 0.01

155
00:07:20,740 --> 00:07:22,160
or something like that.

156
00:07:22,160 --> 00:07:23,875
Anyway, let's do another problem.

157
00:07:23,875 --> 00:07:27,470


158
00:07:27,470 --> 00:07:29,390
Invert colors.

159
00:07:29,389 --> 00:07:30,139
OK.

160
00:07:30,139 --> 00:07:43,899
So let's do the limit as x approaches 0 of

161
00:07:43,899 --> 00:07:50,799
sine of 5x over 2x.

162
00:07:50,800 --> 00:07:54,000
Now this looks a lot like sine of x over x.

163
00:07:54,000 --> 00:07:55,750
And we know sine of x over x.

164
00:07:55,750 --> 00:07:57,279
What is that?

165
00:07:57,279 --> 00:07:59,629
And if you don't believe me, watch the videos where we prove

166
00:07:59,629 --> 00:08:02,519
it using the squeeze theorem.

167
00:08:02,519 --> 00:08:05,129
We proved that the limit as x approaches 0 of sine of

168
00:08:05,129 --> 00:08:09,439
x over x is equal to 1.

169
00:08:09,439 --> 00:08:11,064
Now this looks almost like that.

170
00:08:11,064 --> 00:08:13,165
It would be great if we could get it into that form

171
00:08:13,165 --> 00:08:14,040
and then we'd be done.

172
00:08:14,040 --> 00:08:16,350
So how can we do that?

173
00:08:16,350 --> 00:08:18,000
Well let's try to substitute.

174
00:08:18,000 --> 00:08:20,413
Let's try to get a single variable here instead of

175
00:08:20,413 --> 00:08:21,560
a 5 times a variable.

176
00:08:21,560 --> 00:08:26,610
So let's make a substitution in magenta.

177
00:08:26,610 --> 00:08:30,879
And say that a is equal to 5x.

178
00:08:30,879 --> 00:08:34,439
And then that also means that-- So divide both sides by 5 and

179
00:08:34,440 --> 00:08:37,640
we get x is equal to a over 5.

180
00:08:37,639 --> 00:08:40,049
So let's make that substitution and do this.

181
00:08:40,049 --> 00:08:44,500
So this-- The limit as x approaches 0 would be

182
00:08:44,500 --> 00:08:46,049
the same thing as what?

183
00:08:46,049 --> 00:08:50,079
If approaches-- As x approaches 0 here, a is also going

184
00:08:50,080 --> 00:08:51,720
to approach 0, right?

185
00:08:51,720 --> 00:08:55,450
So this is the same thing as the limit as a approaches 0.

186
00:08:55,450 --> 00:08:57,675
Or you could view it this way: as a approaches 0, x

187
00:08:57,674 --> 00:08:58,719
is still approaching 0.

188
00:08:58,720 --> 00:09:01,490
Maybe at 1/5 the pace, but it's the same thing.

189
00:09:01,490 --> 00:09:09,730
[SIDE COMMENTS]

190
00:09:09,730 --> 00:09:10,149
OK.

191
00:09:10,149 --> 00:09:16,509
Anyway, this should say the limit is a approaches 0, right?

192
00:09:16,509 --> 00:09:18,919
And hopefully you're satisfied that the limit as a approaches

193
00:09:18,919 --> 00:09:21,789
0 is the same-- That x is still approaching 0, right?

194
00:09:21,789 --> 00:09:24,179
Because if a approaches 0 here, that's going

195
00:09:24,179 --> 00:09:25,609
to make x approach 0.

196
00:09:25,610 --> 00:09:27,680
So it's the limit as a approaches 0 of sine

197
00:09:27,679 --> 00:09:34,079
of a over 2 times x.

198
00:09:34,080 --> 00:09:35,770
But x is a over 5.

199
00:09:35,769 --> 00:09:39,490
So it's 2 times a over 5.

200
00:09:39,490 --> 00:09:49,000
And so that is equal to the limit as a approaches 0

201
00:09:49,000 --> 00:09:56,600
of sine of a over 2/5 a.

202
00:09:56,600 --> 00:09:57,659
Well that's the same thing.

203
00:09:57,659 --> 00:09:59,730
We could take this out.

204
00:09:59,730 --> 00:10:01,870
We could take this 1 over 2/5-- right, it's in

205
00:10:01,870 --> 00:10:02,810
the denominator --out.

206
00:10:02,809 --> 00:10:04,239
This is just a constant term, so we can take

207
00:10:04,240 --> 00:10:04,909
it out of the limit.

208
00:10:04,909 --> 00:10:06,409
And obviously if we're taking it out of the denominator,

209
00:10:06,409 --> 00:10:07,189
it flips, right?

210
00:10:07,190 --> 00:10:11,970
Because this is just 1 over 2/5 or 5/2.

211
00:10:11,970 --> 00:10:16,519
So that equals 5/2-- right, I just took this out of the

212
00:10:16,519 --> 00:10:20,230
denominator and it flips, right, because it's 1 over 2/5

213
00:10:20,230 --> 00:10:26,039
--times the limit as a approaches 0 of

214
00:10:26,039 --> 00:10:30,839
sine of a over a.

215
00:10:30,840 --> 00:10:33,980
And now that looks an awful lot like this.

216
00:10:33,980 --> 00:10:35,810
It's just we have an a instead of an x, but that doesn't

217
00:10:35,809 --> 00:10:36,729
make a difference.

218
00:10:36,730 --> 00:10:38,670
So this is equal to 1.

219
00:10:38,669 --> 00:10:45,569
So this whole thing, since this is equal to 1, is equal to 5/2.

220
00:10:45,570 --> 00:10:48,430
And, once again, if you get this answer and you're not so

221
00:10:48,429 --> 00:10:52,179
sure, take your calculator out and try-- calculate.

222
00:10:52,179 --> 00:10:56,750
What is the sine of 5 times 0.001?

223
00:10:56,750 --> 00:11:04,190
So sine of 0.005 divided by 2 times 0.001, so 0.002.

224
00:11:04,190 --> 00:11:06,290
If you take that, you're going to get a number that's awfully

225
00:11:06,289 --> 00:11:07,360
close to this number.

226
00:11:07,360 --> 00:11:08,800
This is approximate.

227
00:11:08,799 --> 00:11:10,179
This is exactly 2.5.

228
00:11:10,179 --> 00:11:12,489
You're probably going to get like 2.49999 or

229
00:11:12,490 --> 00:11:14,690
something like that.

230
00:11:14,690 --> 00:11:15,600
So let's do another one.

231
00:11:15,600 --> 00:11:20,769


232
00:11:20,769 --> 00:11:22,439
OK.

233
00:11:22,440 --> 00:11:30,970
So I have the limit as x approaches 0-- this one looks a

234
00:11:30,970 --> 00:11:33,590
lot like the previous one, although we have some exponents

235
00:11:33,590 --> 00:11:40,820
here --of sine squared of x over x squared.

236
00:11:40,820 --> 00:11:43,040
And that almost looks like sine of x over x, but I

237
00:11:43,039 --> 00:11:44,059
have these squared terms.

238
00:11:44,059 --> 00:11:45,719
What can I do?

239
00:11:45,720 --> 00:11:46,740
Well this is the same thing.

240
00:11:46,740 --> 00:11:49,620
This is sine of x squared over x squared.

241
00:11:49,620 --> 00:11:55,720
So this is the same thing as the limit as x approaches 0

242
00:11:55,720 --> 00:12:03,379
of sine of x over x squared, right?

243
00:12:03,379 --> 00:12:06,230
If you were to take this out, you'd square the numerator,

244
00:12:06,230 --> 00:12:07,480
square the denominator, and you'd get this.

245
00:12:07,480 --> 00:12:10,050
If you took-- If you wanted to change this.

246
00:12:10,049 --> 00:12:11,949
I won't say simplify.

247
00:12:11,950 --> 00:12:13,770
Well now this is interesting.

248
00:12:13,769 --> 00:12:17,179
Well this 2 is-- This is a constant term.

249
00:12:17,179 --> 00:12:18,769
This isn't going to change with the limit.

250
00:12:18,769 --> 00:12:21,129
So essentially, we can say, well this is going to be equal

251
00:12:21,129 --> 00:12:30,049
to the limit as x approaches 0 of sine of x over x, the

252
00:12:30,049 --> 00:12:31,240
whole thing squared.

253
00:12:31,240 --> 00:12:31,769
And how is that?

254
00:12:31,769 --> 00:12:34,279
I couldn't have done this if there was an x here, right,

255
00:12:34,279 --> 00:12:36,000
because that would have changed the limit.

256
00:12:36,000 --> 00:12:37,789
But this is a constant term.

257
00:12:37,789 --> 00:12:42,789
So this-- The base is the only thing that's going to change as

258
00:12:42,789 --> 00:12:44,610
I take-- as x approaches 0.

259
00:12:44,610 --> 00:12:47,870
So I can take the limit within the base, I guess you

260
00:12:47,870 --> 00:12:49,039
could say, and get here.

261
00:12:49,039 --> 00:12:52,299
And now we're back to what we proved in a previous video.

262
00:12:52,299 --> 00:12:54,279
What is the limit as x approaches 0 of

263
00:12:54,279 --> 00:12:55,429
sine of x over x?

264
00:12:55,429 --> 00:12:55,949
Right.

265
00:12:55,950 --> 00:12:59,000
It's 1.

266
00:12:59,000 --> 00:13:00,809
So what's 1 squared?

267
00:13:00,809 --> 00:13:02,349
It is 1.

268
00:13:02,350 --> 00:13:03,740
There you go.

269
00:13:03,740 --> 00:13:05,820
And I've taken 13 minutes of your time.

270
00:13:05,820 --> 00:13:08,330
Hopefully you've found it vaguely Useful.

271
00:13:08,330 --> 00:13:09,000