1
00:00:05,900 --> 00:00:21,900
All right. So what's the first thing we always try? Direct substitution. Over here,

2
00:00:21,900 --> 00:00:35,880
we get 1 over (0 times the square root of 1 plus 0), minus 1 over 0.

3
00:00:35,880 --> 00:00:45,913
0 go to? Well, in limits, it will either go to positive or negative infinity. It

4
00:00:45,913 --> 00:00:46,663
depends

5
00:00:46,540 --> 00:01:06,827
on the left to the right. So let's not worry about the left and right phenomenon. So

6
00:01:06,827 --> 00:01:08,020
this

7
00:01:08,020 --> 00:01:11,678
one over zero goes to either positive or negative infinity. This goes to either

8
00:01:11,678 --> 00:01:12,428
positive or

9
00:01:12,200 --> 00:01:20,960
negative infinity okay and we get this what is infinity minus infinity not zero

10
00:01:22,000 --> 00:01:36,340
we don't know what it is it's another indeterminate form okay if these both go

11
00:01:36,340 --> 00:01:40,540
to positive infinity or sorry let's say the one on the left goes to positive and

12
00:01:40,540 --> 00:01:41,440
this goes to negative.

13
00:01:42,080 --> 00:01:44,780
This would be infinity plus infinity,

14
00:01:44,860 --> 00:01:46,540
which does go to infinity.

15
00:01:47,740 --> 00:01:48,640
But we don't know,

16
00:01:48,540 --> 00:01:50,020
because we didn't do it from the left and the right.

17
00:01:53,160 --> 00:01:55,800
So anytime you get infinity minus infinity,

18
00:01:57,160 --> 00:01:58,800
indeterminate, we gotta do some more work.

19
00:02:01,180 --> 00:02:02,800
All right, so this next trick,

20
00:02:05,460 --> 00:02:08,280
is we just need to combine them into one fraction.

21
00:02:08,860 --> 00:02:11,400
So we get a fraction minus another fraction.

22
00:02:11,460 --> 00:02:13,420
We combine them into one fraction.

23
00:02:13,780 --> 00:02:16,520
So we just need to get a common denominator

24
00:02:17,560 --> 00:02:19,280
of the two fractions.

25
00:02:27,200 --> 00:02:28,100
So we just need to multiply this fraction by something. What do we multiply it by?

26
00:02:34,960 --> 00:02:37,620
So we just need to multiply this guy by something.

27
00:02:37,620 --> 00:02:39,260
What do we multiply this guy by?

28
00:02:41,340 --> 00:02:45,580
I have 1 plus x over 1 plus x.

29
00:02:46,160 --> 00:02:47,880
Now they have a common denominator.

30
00:02:50,820 --> 00:02:59,082
And now if we try direct substitution, we get 1 minus the square root of 1 plus 0,

31
00:02:59,082 --> 00:03:00,000
over 0.

32
00:03:03,720 --> 00:03:06,220
Now we get 0 over 0, which is?

33
00:03:07,100 --> 00:03:09,320
A different indeterminate form.

34
00:03:09,960 --> 00:03:22,680
we're gonna do some more work so now what so we go back to one of our other

35
00:03:22,680 --> 00:03:28,540
tricks multiplied by the conjugate because we got two terms here especially

36
00:03:28,540 --> 00:03:50,281
one involving square root. So let's multiply by 1 plus 1 plus x. And again, the

37
00:03:50,281 --> 00:03:51,640
bottom

38
00:03:51,640 --> 00:03:55,653
or the denominator, which we did not choose the numerator, don't multiply that out.

39
00:03:55,653 --> 00:03:56,403
Just

40
00:03:55,940 --> 00:04:09,361
keep that all factored. Yeah. Again the two middle terms of the conjugate add to be

41
00:04:09,361 --> 00:04:10,200
zero.

42
00:04:10,200 --> 00:04:18,200
So we get 1 plus 0 minus 1 plus x.

43
00:04:21,140 --> 00:04:23,740
Now we distribute this guy.

44
00:04:26,780 --> 00:04:28,640
And now what?

45
00:04:32,820 --> 00:04:37,340
one of our other tricks.

46
00:04:42,000 --> 00:04:45,240
cancel the common factor x.

47
00:04:48,640 --> 00:04:51,280
And now we return to direct substitution and see what happens.

48
00:04:53,940 --> 00:04:59,480
The denominator becomes 1 times 1 plus 1, so the limit is negative one-half.

49
00:05:01,980 --> 00:05:06,660
Okay, so.

50
00:05:06,660 --> 00:05:12,340
So the very first trick was to combine multiple fractions or the sum or

51
00:05:12,340 --> 00:05:14,180
difference of fractions into one fraction.

52
00:05:17,320 --> 00:05:20,480
And then also the other trick is keep trying more tricks.

53
00:05:30,560 --> 00:05:32,340
Another technique.

54
00:05:45,360 --> 00:05:46,760
Alright.

55
00:05:46,760 --> 00:05:47,660
Alright.

56
00:05:50,580 --> 00:05:56,020
Direct substitution: what is sine of zero?

57
00:06:01,480 --> 00:06:17,100
Oh what do we do here okay this next

58
00:06:17,100 --> 00:06:33,760
trick very important one you just memorize the answer memorize okay this

59
00:06:33,760 --> 00:06:38,500
is one of the ones we're just gonna memorize the answer is one here it's not

60
00:06:38,500 --> 00:06:39,400
obvious

61
00:06:42,480 --> 00:06:45,760
Ok, so here is our function sine of x and x.

62
00:06:47,960 --> 00:06:53,735
As you approach zero, these two functions get closer and closer and closer to each

63
00:06:53,735 --> 00:06:54,485
other.

64
00:06:54,120 --> 00:06:57,320
And when you divide two numbers that are very, very close,

65
00:06:58,360 --> 00:07:00,380
it gets closer and closer to 1.

66
00:07:04,480 --> 00:07:09,540
If I zoom in here, the closer you get to 0, the closer

67
00:07:09,540 --> 00:07:11,380
you can tell the difference between the two.

68
00:07:13,300 --> 00:07:18,720
And then if I show you this function, y equals sine x.

69
00:07:31,940 --> 00:07:33,280
It's the green function.

70
00:07:34,840 --> 00:07:37,080
There is a hole right there at x equals zero.

71
00:07:37,260 --> 00:07:39,900
That function doesn't exist at x equals zero.

72
00:07:41,440 --> 00:07:44,780
However, you approach from the left and the right, it gets closer and closer to one.

73
00:07:45,840 --> 00:07:49,000
So the trick is you just memorize this one.

74
00:07:50,360 --> 00:07:51,260
Okay.

75
00:07:53,560 --> 00:07:57,260
Okay. One of the limits we memorize is this one: sine of x over x.

76
00:07:59,240 --> 00:08:00,320
Sine of x over x.

77
00:08:00,320 --> 00:08:02,320
O-Rex...

78
00:08:04,880 --> 00:08:08,040
What about this?

79
00:08:10,160 --> 00:08:12,400
Is this also going to equal one or no?

80
00:08:13,900 --> 00:08:14,800
It's going to be negative.

81
00:08:16,980 --> 00:08:17,880
That's right.

82
00:08:17,860 --> 00:08:20,380
It's just a guess. That's not a bad guess.

83
00:08:21,260 --> 00:08:23,920
Not quite true, but it's not a bad guess.

84
00:08:25,640 --> 00:08:28,480
Is this going to equal one? Why or why not?

85
00:08:31,620 --> 00:08:34,680
Is this true?

86
00:08:34,680 --> 00:08:46,960
wrote down here yes okay right if you had this you take the numerator it's one

87
00:08:46,960 --> 00:08:54,480
multiplied by the reciprocal and you get exactly that okay so this limit is the

88
00:08:54,580 --> 00:09:08,800
limit as x approaches zero of one divided by sine of x over x. And one of the limit

89
00:09:08,800 --> 00:09:21,112
laws that I handed out is the limit of a quotient is the quotient of limits. So we

90
00:09:21,112 --> 00:09:22,480
could write

91
00:09:22,480 --> 00:09:30,760
this as the limit as X approaches 0 of 1 divided by the limit as X approaches 0

92
00:09:31,840 --> 00:09:48,120
sine of X over X and what is this limit? Limit of a constant is just that constant

93
00:09:48,840 --> 00:09:54,340
And this limit we just memorized is 1 is definitely equal to 1.

94
00:09:58,560 --> 00:10:05,300
Long story short, limit as x approaches 0 of sine of x over x or x over sine of x is

95
00:10:05,300 --> 00:10:06,200
1.

96
00:10:09,580 --> 00:10:10,600
Very important.

97
00:10:14,080 --> 00:10:15,780
So if we try direct substitution,

98
00:10:15,960 --> 00:10:20,280
we get 1 minus cosine of 0 over 0.

99
00:10:21,860 --> 00:10:23,420
What is cosine of 0?

100
00:10:25,980 --> 00:10:27,300
0 over 0.

101
00:10:37,420 --> 00:10:39,300
So, is this also going to equal one?

102
00:10:45,600 --> 00:10:47,560
It's going to be one?

103
00:10:48,500 --> 00:10:49,820
Some people shake their head no?

104
00:10:52,520 --> 00:10:56,940
We are also going to memorize this one, but let's figure it out before we get the

105
00:10:56,940 --> 00:10:57,690
answer.

106
00:11:01,100 --> 00:11:02,800
Here's a little trick.

107
00:11:04,800 --> 00:11:09,400
We're going to multiply by 1 plus cosine of x.

108
00:11:12,540 --> 00:11:18,160
which is just a conjugate of the numerator there.

109
00:11:21,520 --> 00:11:28,200
These add to be zero.

110
00:11:31,080 --> 00:11:32,880
Does that help anything?

111
00:11:34,500 --> 00:11:37,320
You direct substitution, you still get 0 over 0.

112
00:11:39,380 --> 00:11:46,160
However, I could substitute this for something.

113
00:11:46,320 --> 00:11:48,380
What is 1 minus cosine squared always equal to?

114
00:11:49,300 --> 00:11:50,200
Sine squared, right?

115
00:11:50,480 --> 00:11:55,000
Because sine squared plus cosine squared equals 1.

116
00:11:58,100 --> 00:12:01,000
So sine squared is one minus cosine squared.

117
00:12:15,640 --> 00:12:18,900
If you try direct substitution, we still get zero over zero.

118
00:12:20,800 --> 00:12:23,240
Any ideas what we could do from here?

119
00:12:26,640 --> 00:12:27,600
What's that?

120
00:12:37,740 --> 00:12:39,780
I'm not sure we could do that.

121
00:12:43,540 --> 00:12:47,760
However, sine squared is sine times sine, correct?

122
00:12:49,120 --> 00:12:52,920
We have an x down here, so we're going to rewrite this as this.

123
00:12:55,840 --> 00:13:09,500
sine of x over x times sine of x over 1 plus cosine of x.

124
00:13:17,060 --> 00:13:19,480
And what is the limit of this guy, guys?

125
00:13:22,260 --> 00:13:26,380
So the limit of a product is the product of limits.

126
00:13:27,360 --> 00:13:28,960
So I'm going to rewrite like this.

127
00:13:32,280 --> 00:13:33,680
this

128
00:13:33,680 --> 00:13:46,000
limit is equal to 1 and now we directly substitute and we get sine of 0 which is

129
00:13:46,000 --> 00:14:03,080
0 1 plus cosine of 0 which is 1 which is 0 over 1 plus 1 which the whole thing is

130
00:14:03,080 --> 00:14:23,105
simply zero. Okay, and what we typically do, or what I typically do, instead of

131
00:14:23,105 --> 00:14:24,440
writing

132
00:14:24,440 --> 00:14:27,725
the limit of the quotient, a limit times a limit, this is technically what's

133
00:14:27,725 --> 00:14:28,475
happening.

134
00:14:27,960 --> 00:14:38,989
what we typically do is just we say this thing goes to 1 and the limit x goes to 0

135
00:14:38,989 --> 00:14:39,739
of

136
00:14:39,540 --> 00:14:47,540
sine of x over 1 plus cosine of x and then we get 0 over 1 or 2 so if you ever

137
00:14:47,540 --> 00:14:53,900
see me like put arrows here that's just meaning whatever this function is it

138
00:14:53,900 --> 00:14:58,800
goes to whatever the limit of it goes to that number just kind of saves you time

139
00:14:58,800 --> 00:15:02,980
from writing the limit times limits on okay so this is another one we memorize

140
00:15:06,980 --> 00:15:10,660
this is really the only other one we memorize

141
00:15:13,900 --> 00:15:15,440
This is just slightly different.

142
00:15:16,020 --> 00:15:17,280
How is this different than 3 to the 1?

143
00:15:23,740 --> 00:15:29,500
The last one was 1 minus cosine of x, this is cosine of x minus 1.

144
00:15:31,880 --> 00:15:33,680
Is this also going to be equal to 0?

145
00:15:44,260 --> 00:15:48,240
Take a guess yes or no everybody give me a thumb.

146
00:15:53,220 --> 00:15:58,260
Okay let's work it out it does equal to one I'm sorry does equal to zero.

147
00:16:01,320 --> 00:16:08,020
If I factor out a negative out of that, I get the limit as x goes to zero.

148
00:16:11,280 --> 00:16:15,707
And then one of our limit laws if we have a coefficient in front what can I do with

149
00:16:15,707 --> 00:16:16,457
that?

150
00:16:18,440 --> 00:16:22,760
You could distribute what are my limit laws here?

151
00:16:28,940 --> 00:16:33,740
Or wasn't at the very top yeah, here it is

152
00:16:37,940 --> 00:16:39,360
This one right here.

153
00:16:40,060 --> 00:16:42,540
You have a coefficient in front, what can I do with the coefficient?

154
00:16:44,600 --> 00:16:47,580
We can bring it outside the limit right here.

155
00:16:55,320 --> 00:16:56,940
So this can come out.

156
00:16:56,940 --> 00:17:09,820
we get negative 1 times this limit and then let me rewrite these two

157
00:17:10,380 --> 00:17:15,980
negative cosine plus x plus 1 is 1 minus cosine of x

158
00:17:20,860 --> 00:17:22,380
and what's this equal to

159
00:17:25,100 --> 00:17:27,120
Zero, one of the ones we memorized.

160
00:17:27,480 --> 00:17:30,960
We just did, so it's negative one times zero, it is zero.

161
00:17:45,500 --> 00:17:50,220
You don't really have to memorize it, but again, the reciprocal of this is the same

162
00:17:50,220 --> 00:17:57,240
value and if these are swapped it's the same value. They're both 0, they're both 1

163
00:18:00,980 --> 00:18:06,100
Let me just write that down anyways.

164
00:18:08,280 --> 00:18:11,140
x over sin x

165
00:18:13,200 --> 00:18:14,540
also equals to

166
00:18:14,540 --> 00:18:16,500
one minute

167
00:18:19,220 --> 00:18:21,800
So the reciprocal has the same value here.

168
00:18:22,340 --> 00:18:23,580
And we can swap that.

169
00:18:23,580 --> 00:18:24,600
I'm going to do some algebra.

170
00:18:39,440 --> 00:18:41,020
All right, next one.

171
00:18:43,800 --> 00:18:47,220
I'll try direct substitution.

172
00:18:47,400 --> 00:18:54,560
We get 0 times secant of 0 times cosecant of 0.

173
00:18:57,140 --> 00:18:59,400
What is secant of zero?

174
00:18:59,800 --> 00:19:04,100
What is cosine of zero?

175
00:19:07,160 --> 00:19:08,800
Which is one.

176
00:19:10,940 --> 00:19:12,140
And cosecant of x?

177
00:19:14,540 --> 00:19:15,540
What is sine of zero?

178
00:19:17,660 --> 00:19:20,100
Zero. So one of this does not exist.

179
00:19:25,160 --> 00:19:26,520
So, we got to do something else.

180
00:19:27,220 --> 00:19:28,220
What are we going to do?

181
00:19:31,360 --> 00:19:32,260
Any ideas?

182
00:19:36,320 --> 00:19:40,100
Figuring out secant, cosecant, cotangent, they are hard, right?

183
00:19:40,460 --> 00:19:42,560
Because then it is just the reciprocal sign goes in.

184
00:19:42,820 --> 00:19:45,400
So the trick is just change everything in terms of sine and cosine.

185
00:19:45,400 --> 00:19:53,180
So all trig functions can be changed in terms of sine and cosine. So this is the

186
00:19:53,180 --> 00:20:05,740
limit as x goes to 0, x times 1 over cosine of x times 1 over sine of x.

187
00:20:11,000 --> 00:20:12,860
Now what?

188
00:20:15,580 --> 00:20:18,340
What does this guy go to as x goes to zero?

189
00:20:19,760 --> 00:20:20,660
One.

190
00:20:20,380 --> 00:20:21,280
One.

191
00:20:21,420 --> 00:20:23,700
So if you want to split it up we can.

192
00:20:28,380 --> 00:20:36,653
The limit as x goes to zero of one over cosine times the limit as x goes to zero of

193
00:20:36,653 --> 00:20:37,480
x over

194
00:20:37,480 --> 00:20:38,680
sine of x.

195
00:20:41,380 --> 00:20:44,160
And then this direct substitution, we get 1.

196
00:20:46,020 --> 00:20:50,200
And this, we memorized as 1, the answer is 1.

197
00:20:54,580 --> 00:20:57,340
So the trick is just change all trig functions

198
00:20:57,340 --> 00:20:58,760
in terms of sine and cosine.

199
00:21:01,700 --> 00:21:09,440
So here it's not sine of x over x, sine of x over 7x.

200
00:21:11,400 --> 00:21:13,020
So what can we do with that?

201
00:21:17,820 --> 00:21:19,420
Yeah, all we do is we factor this is,

202
00:21:20,960 --> 00:21:23,540
it's just a coefficient of like 1 7th.

203
00:21:23,540 --> 00:21:29,140
And we factor it out.

204
00:21:38,700 --> 00:21:41,840
We have 1 7 times 1 is 1 7.

205
00:21:56,820 --> 00:21:59,260
That one shouldn't have been too hard, but let's try this one.

206
00:22:01,900 --> 00:22:04,840
Can I factor out a 5 here?

207
00:22:08,600 --> 00:22:09,500
Definitely not.

208
00:22:10,660 --> 00:22:12,780
You can't factor a number out from inside a trigonometric function.

209
00:22:14,220 --> 00:22:15,220
Uh oh, so now what?

210
00:22:17,900 --> 00:22:19,640
Multiply by 5 over 5.

211
00:22:20,560 --> 00:22:22,300
Okay, I like that.

212
00:22:24,560 --> 00:22:27,420
Multiply this by 5 over 5.

213
00:22:29,380 --> 00:22:33,580
And take this guy out

214
00:22:47,380 --> 00:22:49,240
Why did you want that

215
00:22:53,920 --> 00:22:54,820
It functions.

216
00:22:56,640 --> 00:23:00,300
Yeah, so even though it's not sine of x over x,

217
00:23:01,540 --> 00:23:05,180
it's the same exact thing.

218
00:23:06,900 --> 00:23:11,460
And as x goes to 0, 5x also goes to 0.

219
00:23:13,100 --> 00:23:16,440
And this goes to 0 at exactly the same rate.

220
00:23:17,120 --> 00:23:28,960
If you wanted to, you could let like, uh, u, uh, yeah, let me go, u is five X.

221
00:23:32,920 --> 00:23:38,400
And the limit as X goes to zero of u.

222
00:23:40,300 --> 00:23:41,200
Okay.

223
00:23:40,940 --> 00:23:44,440
This also goes to zero and you end up with something like this.

224
00:23:44,440 --> 00:23:52,140
you don't really have to do this, but sine of u over u.

225
00:23:52,900 --> 00:23:57,280
And this limit, whatever the variable is, is equal to 1.

226
00:24:03,420 --> 00:24:07,980
So the key here, as long as whatever is inside the sine

227
00:24:08,560 --> 00:24:11,800
has to be that same thing down here.

228
00:24:13,540 --> 00:24:16,960
This will equal to, well, in this case, 5.

229
00:24:18,180 --> 00:24:24,380
This will equal, in this case, 5.

230
00:24:24,380 --> 00:24:25,300
bring the letters out.

231
00:24:29,140 --> 00:24:33,720
Oh, here's a good one.

232
00:24:34,000 --> 00:24:34,900
All right.

233
00:24:34,380 --> 00:24:35,280
New topic.

234
00:24:35,140 --> 00:24:36,040
Yeah.

235
00:24:38,520 --> 00:24:46,640
We try direct substitution, we get 3 minus 3, or 3 minus 3, we get 0 over 0.

236
00:24:48,080 --> 00:24:49,980
What in the world are we going to do here?

237
00:24:58,640 --> 00:25:06,580
Uh, this absolute value, what kind of function is that?

238
00:25:08,460 --> 00:25:15,398
What's that? It's a piecewise function. Okay, let's look at the definition of the

239
00:25:15,398 --> 00:25:17,380
absolute value of x

240
00:25:17,380 --> 00:25:18,792
minus three. This is a piecewise function. It's either going to be x minus three or

241
00:25:18,792 --> 00:25:19,542
it's

242
00:25:18,880 --> 00:25:26,000
x minus three if x minus three is greater than or equal to zero and if x

243
00:25:26,000 --> 00:25:41,700
minus three is less than zero so x minus three it's either x minus three if x is

244
00:25:41,700 --> 00:25:43,500
Yeah, greater than or equal to zero.

245
00:25:44,620 --> 00:25:46,680
Sorry, greater than or equal to three.

246
00:25:49,320 --> 00:25:51,060
Negative x plus three.

247
00:25:54,340 --> 00:25:56,780
Let me just leave this x minus three.

248
00:26:00,240 --> 00:26:01,800
If x is less than three.

249
00:26:03,360 --> 00:26:05,040
Okay, all absolute value functions

250
00:26:05,040 --> 00:26:06,520
are piecewise functions, right?

251
00:26:06,740 --> 00:26:07,800
They all, they look like a V.

252
00:26:08,580 --> 00:26:09,900
There's two different pieces.

253
00:26:15,000 --> 00:26:21,080
Okay, so going back here, we're going to have to write a piecewise definition.

254
00:26:27,620 --> 00:26:29,940
So we're approaching three here, right?

255
00:26:30,060 --> 00:26:31,700
Not from left to right, but both sides.

256
00:26:33,500 --> 00:26:36,480
But this thing depends on each side of the three.

257
00:26:36,480 --> 00:26:42,311
So we're gonna have to break these up into the left-hand limit and the right-hand

258
00:26:42,311 --> 00:26:43,061
limit.

259
00:26:43,160 --> 00:26:54,440
So the limit as x approaches 3.

260
00:26:54,900 --> 00:26:55,860
Start from the left.

261
00:27:06,860 --> 00:27:14,193
OK, so since we're going from the left, are we always greater than or equal to 3 or

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00:27:14,193 --> 00:27:14,943
less

263
00:27:14,600 --> 00:27:15,500
than 3?

264
00:27:16,600 --> 00:27:18,380
Less than 3.

265
00:27:19,280 --> 00:27:22,360
So I'm going to substitute this for this right here.

266
00:27:23,500 --> 00:27:29,500
So this is the limit as x approaches 3 from the left

267
00:27:30,920 --> 00:27:37,720
of negative, open parenthesis, x minus 3, close parenthesis, over 3 minus x.

268
00:27:44,460 --> 00:27:48,660
Which is the limit as x approaches 3 from the left.

269
00:27:48,660 --> 00:27:59,124
I distribute that negative and switch it, we get 3 minus x over 3 minus x, which

270
00:27:59,124 --> 00:27:59,874
simplifies

271
00:27:59,740 --> 00:28:01,880
to 1.

272
00:28:04,560 --> 00:28:11,980
And now let's take the limit as x approaches 3 but from the right-hand side.

273
00:28:15,040 --> 00:28:21,120
Now, since we're approaching three from the right, this absolute value.

274
00:28:26,460 --> 00:28:31,220
Okay, we're to the right of three, so I can substitute this in.

275
00:28:33,860 --> 00:28:38,720
So this is x minus three over three minus x.

276
00:28:41,280 --> 00:28:42,320
And...

277
00:28:46,940 --> 00:28:49,020
How do I simplify this?

278
00:28:53,100 --> 00:28:54,060
You can factor out the negative.

279
00:28:57,920 --> 00:29:09,780
Either from the numerator or denominator, we get negative x plus three, which if you

280
00:29:09,780 --> 00:29:19,640
around they simplify to 1 and now we get negative so the limit from the left

281
00:29:19,640 --> 00:29:26,480
exists it's equal to 1 the limit from the right exists negative 1 so what is

282
00:29:26,480 --> 00:29:35,840
the limit as X approaches 3 this does not exist

283
00:29:46,220 --> 00:29:51,886
Just to get an idea of this function right here, I think it looks something like

284
00:29:51,886 --> 00:29:52,636
this.

285
00:29:54,300 --> 00:29:57,280
everywhere to the right.

286
00:30:00,760 --> 00:30:02,440
It's got a cool function.

287
00:30:04,500 --> 00:30:09,161
We're to the right of 3, it's always going to be negative 1, or left to be positive

288
00:30:09,161 --> 00:30:09,911
1.

289
00:30:10,640 --> 00:30:11,740
Okay, so what's this trick?

290
00:30:12,640 --> 00:30:16,020
Is we have to break up piecewise functions into two pieces.

291
00:30:18,200 --> 00:30:20,540
And especially involving absolute value.

292
00:30:20,540 --> 00:30:24,320
Okay, so break up.

293
00:30:27,060 --> 00:30:44,387
Okay, especially involving anything involving absolute values is a piecewise

294
00:30:44,387 --> 00:30:46,120
function.

295
00:30:48,740 --> 00:30:49,960
All right, what are we going to do here?

296
00:30:53,720 --> 00:30:58,660
Actually, let's do a, no, let's just do this one.

297
00:31:02,300 --> 00:31:05,180
Here we cannot just use direct substitution.

298
00:31:07,220 --> 00:31:13,660
If I use direct substitution, you plug in 1 to the function.

299
00:31:13,820 --> 00:31:14,860
What is f of 1?

300
00:31:18,640 --> 00:31:21,720
Do I plug it into here or do I plug it into here?

301
00:31:23,100 --> 00:31:26,160
The top one, because that's in this domain right here.

302
00:31:27,100 --> 00:31:32,080
So that's ln of 1, which is equal to 0.

303
00:31:34,220 --> 00:31:40,399
We can't use direct substitution because, well we can only if we know it's

304
00:31:40,399 --> 00:31:41,149
continuous.

305
00:31:41,780 --> 00:31:45,457
And we don't know if this function is continuous because this has two different

306
00:31:45,457 --> 00:31:46,207
pieces

307
00:31:46,480 --> 00:31:52,580
Okay, so especially any time it changes right here at whatever number this is

308
00:31:52,580 --> 00:31:56,755
We got to just break up and do the left-hand limit and the right-hand limit. So

309
00:31:56,755 --> 00:31:58,060
let's start with the left

310
00:31:59,120 --> 00:32:00,080
X approaches

311
00:32:01,240 --> 00:32:03,080
0 sorry 1

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00:32:04,480 --> 00:32:05,720
From the left

313
00:32:18,980 --> 00:32:24,260
Now, since we're always to the left, what is f of x?

314
00:32:27,100 --> 00:32:28,160
Is x squared.

315
00:32:28,440 --> 00:32:33,620
So I can write the limit as x approaches 1 from the left of x squared.

316
00:32:36,220 --> 00:32:38,480
Now this is a continuous function.

317
00:32:38,480 --> 00:32:41,160
So we can just use direct substitution.

318
00:32:42,280 --> 00:32:44,860
If we plug it in, we get 1 squared, which is 1.

319
00:32:50,960 --> 00:32:56,660
As we approach from the

320
00:32:56,660 --> 00:33:01,260
right of f of x

321
00:33:10,280 --> 00:33:17,000
Okay, since we're always to the right of one, our function is equal to ln of x.

322
00:33:18,520 --> 00:33:22,040
So I can substitute in for f of x ln of x.

323
00:33:25,200 --> 00:33:28,880
And again, this is a continuous function, so we can just plug it in.

324
00:33:33,180 --> 00:33:40,340
And we get ln of 1 and ln of 1 is 0.

325
00:33:44,300 --> 00:33:47,480
So the left-hand limit exists, the right-hand limit exists.

326
00:33:49,720 --> 00:33:52,660
So what is this limit as x approaches 1?

327
00:33:53,800 --> 00:34:01,040
It does not exist because the left-hand limit does not equal the right-hand limit.

328
00:34:07,580 --> 00:34:17,620
A more interesting question is: what if the lower piece were x squared plus c?

329
00:34:17,620 --> 00:34:29,060
The question is, what value of C makes this continuous everywhere?

330
00:34:29,060 --> 00:34:29,960
everywhere.

331
00:34:32,340 --> 00:34:33,860
So you look at each piece.

332
00:34:34,200 --> 00:34:35,400
Is this continuous everywhere?

333
00:34:38,320 --> 00:34:39,220
Not really.

334
00:34:40,140 --> 00:34:41,820
It is continuous on what domain?

335
00:34:42,260 --> 00:34:43,440
Why won't it change colors?

336
00:34:44,820 --> 00:34:47,440
As long as x is greater than zero, it's continuous.

337
00:34:48,160 --> 00:34:50,860
And it's only this when it's continuous, or sorry,

338
00:34:51,220 --> 00:34:52,120
when it's that function.

339
00:34:52,320 --> 00:34:54,300
So this is continuous everywhere here.

340
00:34:55,660 --> 00:34:58,200
This is, no matter what c is, is a polynomial.

341
00:34:59,120 --> 00:35:00,860
So it's definitely continuous everywhere here.

342
00:35:01,060 --> 00:35:05,940
So the only question is, at 1, we got to make it continuous.

343
00:35:07,560 --> 00:35:16,040
So we got to, this is.

344
00:35:21,520 --> 00:35:22,900
This is natural log.

345
00:35:24,220 --> 00:35:29,160
And then this is just x squared.

346
00:35:33,220 --> 00:35:37,280
So we just have to shift it up or down so at one they're exactly the same value.

347
00:35:40,560 --> 00:35:44,860
Long story short, we just need to have the two limits equal.

348
00:35:47,360 --> 00:35:55,260
With plus c, the left-hand limit would be 1 plus c.

349
00:35:57,060 --> 00:35:58,700
So to make it continuous, we just

350
00:35:58,700 --> 00:36:00,780
have to make sure these match.

351
00:36:02,100 --> 00:36:04,040
So 1 plus C equals 0.

352
00:36:04,720 --> 00:36:05,840
C is negative 1.

353
00:36:09,720 --> 00:36:11,940
OK, that was, we're going to get to more

354
00:36:11,940 --> 00:36:13,180
about continuous functions later,

355
00:36:13,180 --> 00:36:18,620
but just a little quick question. Let's see.