1
00:00:04,180 --> 00:00:11,900
Okay, so today we're talking about limits.

2
00:00:12,460 --> 00:00:13,520
What is a limit?

3
00:00:17,040 --> 00:00:21,840
this okay you don't have to write this

4
00:00:23,720 --> 00:00:26,840
first sentence down actually you don't really you don't need to write this down

5
00:00:26,840 --> 00:00:34,980
yet so I put it in there my calculator some function you don't know what it is

6
00:00:36,160 --> 00:00:42,320
secret I put into y1 and here's what I want to know I want to know what is our

7
00:00:42,320 --> 00:00:55,520
function approaching as X approaches the number 2 okay I don't know what it looks

8
00:00:55,520 --> 00:01:09,980
like but as as we plug in values closer and closer to X what are these values

9
00:01:09,980 --> 00:01:17,378
getting close to? What is this number? Okay, and as we approach from the other side,

10
00:01:17,378 --> 00:01:18,128
what

11
00:01:17,840 --> 00:01:26,100
is this approaching? Okay, so this is the whole idea of a limit is as x gets closer

12
00:01:26,100 --> 00:01:36,302
to some number, what is our function value getting closer to? Okay, and this is the

13
00:01:36,302 --> 00:01:37,052
notation

14
00:01:36,940 --> 00:01:43,528
we use. LIM stands for limit. And this is x, we have a little arrow always

15
00:01:43,528 --> 00:01:44,278
approaches

16
00:01:43,940 --> 00:01:49,700
some number, we put a number here. And sometimes you'll see a positive or a negative

17
00:01:49,700 --> 00:01:50,450
sign,

18
00:01:50,060 --> 00:01:53,500
and also so that means, and of this function.

19
00:01:54,040 --> 00:01:56,620
So we wanna know what x approaches two,

20
00:01:57,400 --> 00:01:58,300
as x approaches two,

21
00:01:58,320 --> 00:02:01,440
so what numbers should I plug into my calculator?

22
00:02:09,660 --> 00:02:11,180
Kind of like we did on the quiz.

23
00:02:12,020 --> 00:02:13,880
We kept plugging in numbers closer and closer to 4.

24
00:02:14,560 --> 00:02:17,500
So let's try like 1.9.

25
00:02:21,620 --> 00:02:22,660
1.99.

26
00:02:26,240 --> 00:02:27,200
1.999.

27
00:02:47,820 --> 00:02:52,620
So I'm plugging in closer x values closer and closer to 2, right?

28
00:02:53,260 --> 00:02:56,400
And what does it look like after x is getting closer and closer and closer to 2?

29
00:02:56,400 --> 00:02:57,420
7.

30
00:02:58,840 --> 00:02:59,740
OK.

31
00:02:59,660 --> 00:03:10,540
So whatever this function is, we think

32
00:03:10,540 --> 00:03:14,960
this is getting closer and closer to 7,

33
00:03:15,420 --> 00:03:17,460
even though we do not know what happens at x = 2.

34
00:03:18,460 --> 00:03:20,340
For the limit, we do not care what happens exactly at x = 2.

35
00:03:21,180 --> 00:03:23,800
All we care is what's happening as we get closer to it.

36
00:03:23,800 --> 00:03:24,700
So what we actually just found, where's my cap?

37
00:03:24,220 --> 00:03:26,760
We actually only approached it from the left side,

38
00:03:27,520 --> 00:03:29,060
numbers less than two.

39
00:03:30,000 --> 00:03:33,240
Okay, we also need to approach it from the right side,

40
00:03:33,860 --> 00:03:34,920
numbers greater than two.

41
00:03:35,320 --> 00:03:36,220
So I'm gonna plug in, oops.

42
00:03:35,840 --> 00:03:42,016
Okay, so as we plug the numbers from the right, the numbers are slightly greater

43
00:03:42,016 --> 00:03:42,840
than two

44
00:03:42,840 --> 00:03:47,847
and the closer we got to two, what is our function getting closer and closer to

45
00:03:47,847 --> 00:03:48,597
seven

46
00:03:48,940 --> 00:03:49,840
as well?

47
00:03:51,260 --> 00:03:57,054
Okay, limits, we don't care what happens at this number two, we only care what's

48
00:03:57,054 --> 00:03:57,804
happening

49
00:03:57,440 --> 00:03:58,700
from the left and from the right.

50
00:04:01,200 --> 00:04:09,500
Okay, so whatever this function is, and so we use certain notation.

51
00:04:09,980 --> 00:04:15,722
If we approach from the left, we put a little, it looks like to the power of a

52
00:04:15,722 --> 00:04:16,472
negative sign.

53
00:04:18,460 --> 00:04:24,544
Okay, this is me, we're approaching x is approaching 2, but number is less than 2 of

54
00:04:24,544 --> 00:04:25,294
this function.

55
00:04:28,420 --> 00:04:30,620
Okay, and the answer was 7.

56
00:04:33,100 --> 00:04:40,994
If we approach from the right side, it's called the right-hand limit, this was also

57
00:04:40,994 --> 00:04:41,744
7.

58
00:04:42,720 --> 00:04:47,878
Okay, so limits, all we care about is what's happening as x gets closer to some

59
00:04:47,878 --> 00:04:48,628
number,

60
00:04:49,460 --> 00:04:51,320
both from the right and from the left.

61
00:04:52,400 --> 00:05:00,005
If these both are the same number, this limit, which is just approaching 2, not from

62
00:05:00,005 --> 00:05:00,755
the

63
00:05:00,480 --> 00:05:04,220
left, not from the right, but both sides, is equal to 7 as well.

64
00:05:05,220 --> 00:05:06,800
So this is called a left-hand limit.

65
00:05:11,140 --> 00:05:16,280
It's called the right-hand limit.

66
00:05:16,280 --> 00:05:34,313
limit, okay, and if the limit as x approaches whatever number this is, from the left

67
00:05:34,313 --> 00:05:35,440
equals

68
00:05:35,440 --> 00:05:53,139
the limit as x approaches a from the right, then the limit as x approaches a, which

69
00:05:53,139 --> 00:05:54,180
is

70
00:05:54,180 --> 00:05:55,540
from the left and right.

71
00:05:58,840 --> 00:05:59,611
Okay, so if these are both some number let's call it L the limit then this is also

72
00:05:59,611 --> 00:06:00,361
equal to L

73
00:05:59,700 --> 00:06:02,740
And we do not care what f(2) is.

74
00:06:04,800 --> 00:06:10,440
And we find limits we don't care what f of that of this value is. All we care

75
00:06:10,440 --> 00:06:23,680
about is what's happening from the left and from the right. In fact, what is f of 2

76
00:06:23,680 --> 00:06:30,204
here? It does not even exist. That is okay—we do not care about f(2) while finding

77
00:06:30,204 --> 00:06:31,020
this limit.

78
00:06:31,020 --> 00:06:35,707
Later, we will care because the function value helps tell us whether the function is

79
00:06:35,707 --> 00:06:36,457
continuous.

80
00:06:36,020 --> 00:06:42,140
For now, one way to find a limit is to plug in values just to the left and right.

81
00:06:42,140 --> 00:06:46,000
we're finding limits one way is this way you keep plugging in values if you're

82
00:06:46,000 --> 00:06:46,900
we're finding limits one way is this way you keep plugging in values if you're

83
00:06:46,300 --> 00:06:51,000
given the equations just the left of it just the right of it and they keep

84
00:06:51,000 --> 00:06:58,920
getting closer to some number that is the limit what was this secret function

85
00:06:58,920 --> 00:07:00,100
I think I know.

86
00:07:01,520 --> 00:07:02,420
No idea.

87
00:07:05,100 --> 00:07:06,460
Let's see what it is.

88
00:07:18,420 --> 00:07:20,640
But what kind of function was this?

89
00:07:21,320 --> 00:07:22,300
What do you call this kind of function?

90
00:07:25,520 --> 00:07:26,420
Rational!

91
00:07:27,740 --> 00:07:28,640
Thank you!

92
00:07:31,640 --> 00:07:32,640
What is a rational function?

93
00:07:36,680 --> 00:07:45,140
A rational function is a ratio of two polynomials.

94
00:07:55,080 --> 00:08:05,440
For example, x squared over e to the x is not a rational function. It is not just a

95
00:08:05,440 --> 00:08:13,600
ratio of functions or expressions. It must be a ratio of two

96
00:08:13,600 --> 00:08:18,200
polynomials. Thank you. As long as both are polynomials, it is called a

97
00:08:18,200 --> 00:08:24,200
rational function. As a review, rational functions have one of two

98
00:08:24,200 --> 00:08:31,880
things they either have a vertical asymptotes or what's the other option

99
00:08:37,720 --> 00:08:44,680
Oh, well, where is this?

100
00:08:49,580 --> 00:08:55,283
Okay, so when values that make the denominator zero will either be vertical

101
00:08:55,283 --> 00:08:56,160
asymptote or

102
00:08:56,160 --> 00:08:57,980
a hole in the graph.

103
00:08:58,180 --> 00:08:59,080
Thank you.

104
00:08:59,560 --> 00:09:00,460
Okay.

105
00:09:01,520 --> 00:09:05,700
Does this function have a hole in the graph or a vertical asymptote?

106
00:09:06,640 --> 00:09:07,580
A hole.

107
00:09:07,580 --> 00:09:08,480
A hole.

108
00:09:09,000 --> 00:09:09,900
Okay.

109
00:09:09,960 --> 00:09:13,060
At two, it doesn't look like this.

110
00:09:13,480 --> 00:09:23,120
It looks like a hole in the graph.

111
00:09:24,880 --> 00:09:26,920
How can we tell from here that it has a hole?

112
00:09:30,320 --> 00:09:31,600
That's a hole in the graph.

113
00:09:32,700 --> 00:09:35,060
If this wasn't here, it would be a vertical asymptote.

114
00:09:37,000 --> 00:09:42,760
What would happen, in fact, let me put that in the calculator.

115
00:09:54,680 --> 00:10:00,360
OK, so now it no longer has a hole.

116
00:10:01,280 --> 00:10:03,860
There's going to be a vertical asymptote at x equals 2.

117
00:10:07,720 --> 00:10:15,040
What's going to happen when I plug in numbers close to two now?

118
00:10:17,820 --> 00:10:18,720
Any guesses?

119
00:10:37,900 --> 00:10:39,760
Okay, what's happening with the y-value?

120
00:10:43,960 --> 00:10:47,693
Just keeps getting larger except it's negative, so it's really getting smaller and

121
00:10:47,693 --> 00:10:48,443
smaller

122
00:10:47,980 --> 00:10:49,520
smaller and smaller and smaller.

123
00:10:50,600 --> 00:10:58,020
Okay, in fact, and then what's if I plug in numbers greater than two?

124
00:11:07,320 --> 00:11:11,180
I like 2.01.

125
00:11:14,600 --> 00:11:17,400
2.001.

126
00:11:21,360 --> 00:11:24,000
2.001.

127
00:11:25,460 --> 00:11:27,280
It just keeps getting larger and larger and larger.

128
00:11:27,980 --> 00:11:30,080
OK, so if this happens on the calculator,

129
00:11:30,220 --> 00:11:33,840
if you're given a secret function,

130
00:11:35,300 --> 00:11:36,200
you're plugging it back.

131
00:11:36,240 --> 00:11:37,480
Or you could plug in the number.

132
00:11:37,620 --> 00:11:38,520
It doesn't matter.

133
00:11:39,380 --> 00:11:41,000
Let's talk about that limit, actually.

134
00:11:42,360 --> 00:11:43,600
Ch-ch-ch-ch.

135
00:11:50,640 --> 00:11:52,280
So this is my function.

136
00:11:53,960 --> 00:11:55,660
Was it x squared plus 3?

137
00:11:56,180 --> 00:11:57,080
No—minus 3.

138
00:11:56,660 --> 00:11:58,000
So the function is (x squared minus 3) divided by (x minus 2).

139
00:11:58,340 --> 00:12:03,920
Okay, and I can either write f of x here or I can write in the whole thing

140
00:12:07,140 --> 00:12:10,780
Okay, so we plug in values they kept going

141
00:12:11,620 --> 00:12:13,800
Smaller or smaller so we say

142
00:12:16,880 --> 00:12:18,280
The left-hand limit is negative infinity.

143
00:12:18,280 --> 00:12:32,500
Some people say the limit does not exist here, but for this one-sided limit we write

144
00:12:32,500 --> 00:12:40,140
negative infinity. Now, what is the two-sided limit

145
00:12:40,140 --> 00:12:54,700
as x approaches 2 of this function. If this does not equal this, it does not exist.

146
00:12:55,040 --> 00:13:00,480
So today we're only talking about limits based on graphs of a function or based on

147
00:13:00,480 --> 00:13:01,230
tables

148
00:13:00,820 --> 00:13:06,440
of values. The last example is table values get plugging in numbers or you're

149
00:13:06,440 --> 00:13:17,180
given the tables of the X and the Y values. Friday we'll start learning based

150
00:13:17,180 --> 00:13:23,360
on equation or yeah function equations. Alright first question what is the limit

151
00:13:24,160 --> 00:13:25,860
or let me write down here.

152
00:13:27,400 --> 00:13:37,120
The limit as x approaches 2 from the left g of x.

153
00:13:43,240 --> 00:13:49,208
So as x gets closer and closer to 2 from the left, what is this getting closer and

154
00:13:49,208 --> 00:13:49,958
closer

155
00:13:49,540 --> 00:13:50,440
and closer to?

156
00:13:50,800 --> 00:13:51,700
3.

157
00:13:51,420 --> 00:14:07,980
so this limit equals 3 the limit as X approaches 2 from the right as X gets

158
00:14:08,180 --> 00:14:25,342
closer and closer to f of x gets closer and closer to 1. And what is the limit as x

159
00:14:25,342 --> 00:14:26,200
approaches

160
00:14:26,200 --> 00:14:35,418
2? It does not exist. We don't have to look at anything except this number does not

161
00:14:35,418 --> 00:14:36,168
equal

162
00:14:35,960 --> 00:14:40,700
this number, the left-hand limit does not equal the right-hand limit, the limit as x

163
00:14:40,700 --> 00:14:44,380
approaches 2 does not exist.

164
00:14:48,840 --> 00:14:51,720
And what is g of 2?

165
00:14:57,100 --> 00:14:58,780
Where is the red dot at 2?

166
00:14:59,820 --> 00:15:00,800
There is none.

167
00:15:00,800 --> 00:15:03,500
So this does not exist as well.

168
00:15:03,540 --> 00:15:06,620
So at every x value, we always have a left-hand limit,

169
00:15:06,840 --> 00:15:09,080
a right-hand limit, the limit,

170
00:15:10,420 --> 00:15:11,920
and the actual function value.

171
00:15:13,480 --> 00:15:16,320
And they all might be a number, they all might not exist.

172
00:15:16,680 --> 00:15:20,160
So let's look at five here.

173
00:15:35,260 --> 00:15:42,840
The limit as x approaches five from the left.

174
00:15:48,400 --> 00:15:53,720
As x approaches 5 from the left, these values get closer and closer to 2.

175
00:15:58,120 --> 00:16:04,360
This is the limit as x approaches 5 from the right.

176
00:16:06,900 --> 00:16:14,360
as x approaches 5 from the right or g of x approaches 2.

177
00:16:19,860 --> 00:16:22,680
So what is the limit as x approaches 2?

178
00:16:24,860 --> 00:16:25,760
5?

179
00:16:25,520 --> 00:16:26,420
1.

180
00:16:27,340 --> 00:16:28,240
This?

181
00:16:28,020 --> 00:16:28,920
2.

182
00:16:28,780 --> 00:16:29,680
Sorry, 2.

183
00:16:32,000 --> 00:16:33,020
My bad.

184
00:16:34,980 --> 00:16:37,540
Five. My bad.

185
00:16:39,580 --> 00:16:41,340
Since the left-hand limit

186
00:16:41,340 --> 00:16:44,780
equals the right-hand limit, so the two-sided limit is exactly 2.

187
00:16:49,240 --> 00:16:51,480
And what is G of five?

188
00:16:53,440 --> 00:16:54,340
One.

189
00:16:56,120 --> 00:16:57,020
Okay.

190
00:16:57,960 --> 00:17:00,820
Is our function discontinuous at 5?

191
00:17:02,720 --> 00:17:03,620
Yeah, definitely.

192
00:17:05,620 --> 00:17:09,520
OK. And how can we tell?

193
00:17:09,900 --> 00:17:14,280
Well, obviously we look at it. But if all I gave you is this information,

194
00:17:14,440 --> 00:17:19,520
the way we tell if it's continuous is does this equal the limit?

195
00:17:21,500 --> 00:17:25,600
OK, if the function value equals the limit,

196
00:17:25,600 --> 00:17:29,613
the function is continuous. If the function value does not equal the limit, it is

197
00:17:29,613 --> 00:17:30,363
discontinuous.

198
00:17:31,300 --> 00:17:33,000
And then our previous example

199
00:17:33,980 --> 00:17:37,377
In our previous example, the function value did not exist, so it was definitely

200
00:17:37,377 --> 00:17:38,127
discontinuous.

201
00:17:38,340 --> 00:17:41,480
But this doesn't equal well this doesn't exist either, so

202
00:17:43,900 --> 00:17:46,900
But in fact that's going to be the definition of

203
00:17:47,500 --> 00:17:54,160
A function is continuous at a point when the limit equals the function value.

204
00:17:54,160 --> 00:17:55,900
We'll define that later.

205
00:17:55,940 --> 00:17:58,200
Okay, let's do another one.

206
00:18:05,760 --> 00:18:12,320
What is the limit as x approaches 0 from the left?

207
00:18:18,840 --> 00:18:26,800
So as x approaches zero, f of x gets closer and closer to, let's call it 2.5.

208
00:18:32,880 --> 00:18:43,067
The limit as x approaches zero from the right, as x approaches zero from the right,

209
00:18:43,067 --> 00:18:44,340
f of

210
00:18:44,340 --> 00:18:48,060
x gets closer and closer to 2.5.

211
00:18:50,500 --> 00:18:54,340
What is the limit as x approaches zero?

212
00:18:54,340 --> 00:19:00,740
2.5. 2.5.

213
00:19:02,560 --> 00:19:07,020
And what is g of 0? 2.5.

214
00:19:07,420 --> 00:19:11,920
2.5. Is it continuous at 0?

215
00:19:13,140 --> 00:19:14,040
Yeah. Okay.

216
00:19:17,160 --> 00:19:20,360
And in fact, if we know the function is continuous,

217
00:19:21,200 --> 00:19:26,560
Let's say a 2 and 5 are its only discontinuities.

218
00:19:32,080 --> 00:19:34,480
Or let me ask you this.

219
00:19:34,480 --> 00:19:38,280
the limit as x approaches zero,

220
00:19:45,360 --> 00:19:49,020
x to the fifth minus seven x plus four.

221
00:19:53,120 --> 00:19:54,580
Is this a continuous function?

222
00:19:59,380 --> 00:20:00,980
What do we call this type of function?

223
00:20:02,420 --> 00:20:03,320
Polynomial.

224
00:20:03,200 --> 00:20:04,220
A polynomial.

225
00:20:05,320 --> 00:20:06,220
Okay.

226
00:20:06,400 --> 00:20:09,760
All polynomials are continuous over all real numbers.

227
00:20:10,860 --> 00:20:11,760
Okay.

228
00:20:12,580 --> 00:20:19,375
Since we know it's continuous over all real numbers, we know this limit exactly

229
00:20:19,375 --> 00:20:20,125
equals

230
00:20:19,860 --> 00:20:21,640
whatever f of zero is.

231
00:20:22,240 --> 00:20:23,240
Let's call this f.

232
00:20:24,880 --> 00:20:25,900
And what is f of zero?

233
00:20:26,580 --> 00:20:28,260
It will be four.

234
00:20:29,520 --> 00:20:30,420
Okay.

235
00:20:29,940 --> 00:20:34,340
So if a function is continuous at a value—or over all values—

236
00:20:34,340 --> 00:20:39,840
the limit as x approaches that value equals the function value.

237
00:20:40,860 --> 00:20:42,220
okay, so

238
00:20:45,820 --> 00:20:48,420
Like at four here

239
00:20:50,940 --> 00:20:57,760
It's definitely continuous there so the limit as X approaches four from the left

240
00:20:59,980 --> 00:21:04,040
is going to equal the limit as x approaches 4 from the right.

241
00:21:07,320 --> 00:21:10,800
It's going to equal the limit as x approaches 4.

242
00:21:14,500 --> 00:21:17,180
It's going to equal g of 4, whatever that number is.

243
00:21:18,460 --> 00:21:21,260
Let's call it 1.8. I'm just making up a number.

244
00:21:22,760 --> 00:21:26,360
If the function is continuous, the left-hand limit equals the right-hand limit,

245
00:21:26,360 --> 00:21:28,060
the limit equals the function value.

246
00:21:31,220 --> 00:21:34,740
So every other number here, besides two and five,

247
00:21:34,920 --> 00:21:36,760
the limit from the left will equal the limit from the right.

248
00:21:37,180 --> 00:21:40,100
The limit will equal the function value.

249
00:21:41,220 --> 00:21:42,120
Make sense so far?

250
00:21:42,820 --> 00:21:44,000
Okay, if you're getting a graph, it's real easy.

251
00:21:44,040 --> 00:21:48,020
Okay, in fact, actually let's do this example.

252
00:21:51,920 --> 00:21:54,960
What is the limit as x approaches, let's go to two.

253
00:21:55,280 --> 00:21:56,180
Oh no, let's go one.

254
00:21:56,060 --> 00:22:02,420
The limit as x approaches 1 from the left is 1.

255
00:22:03,280 --> 00:22:04,640
It keeps getting closer and closer to 1.

256
00:22:05,280 --> 00:22:08,280
As x approaches 1 from the right, 1.

257
00:22:09,240 --> 00:22:11,180
What is the limit as x approaches 1?

258
00:22:12,200 --> 00:22:13,100
1.

259
00:22:13,280 --> 00:22:14,700
What is f of 1?

260
00:22:15,400 --> 00:22:16,300
1.

261
00:22:16,560 --> 00:22:17,460
No.

262
00:22:18,460 --> 00:22:19,360
Ok.

263
00:22:19,900 --> 00:22:22,100
We have a hole in the graph right here.

264
00:22:23,680 --> 00:22:29,233
You can never tell on Desmos or the graphing calculator there's a hole so be real

265
00:22:29,233 --> 00:22:29,983
careful

266
00:22:30,720 --> 00:22:35,220
Okay, because it's so small no matter how far I zoom in here

267
00:22:36,220 --> 00:22:37,520
Where is it?

268
00:22:38,080 --> 00:22:40,100
You will never see the hole

269
00:22:41,240 --> 00:22:43,620
Okay, so be real careful on

270
00:22:44,720 --> 00:22:46,340
graphing calculators on Desmos

271
00:22:46,900 --> 00:22:48,900
You will never see holes in the graph

272
00:22:50,200 --> 00:22:52,900
Okay, that's why when we draw them

273
00:22:53,960 --> 00:23:02,500
Like here we make the whole giant so, you know, it's a whole that makes sense. Okay

274
00:23:03,440 --> 00:23:05,280
You don't have to draw this

275
00:23:06,200 --> 00:23:07,340
We'll just do it real quick.

276
00:23:13,140 --> 00:23:19,740
What is the limit as this is 0, 0, right?

277
00:23:20,900 --> 00:23:25,120
As x approaches 0 from the left, let's call this f.

278
00:23:28,580 --> 00:23:31,960
As x approaches 0 from the left, f of x approaches?

279
00:23:33,320 --> 00:23:34,220
Maybe 1.

280
00:23:33,800 --> 00:23:39,360
What is the limit as x approaches 0 from the right?

281
00:23:42,240 --> 00:23:45,920
As x approaches 0, this approaches negative 2.

282
00:23:47,840 --> 00:23:49,660
So both of these limits exist.

283
00:23:50,360 --> 00:23:51,820
Anytime you get a number it exists.

284
00:23:53,380 --> 00:23:57,340
And what is the limit as x approaches 0?

285
00:23:59,020 --> 00:24:02,040
It does not exist.

286
00:24:02,980 --> 00:24:03,880
Why?

287
00:24:03,260 --> 00:24:09,489
because the left-hand limit does not equal the right-hand limit. Okay and what is F

288
00:24:09,489 --> 00:24:10,320
of 0?

289
00:24:12,760 --> 00:24:21,400
negative 1. Okay is it discontinuous at X equals 0? Definitely, that's some sort of

290
00:24:21,400 --> 00:24:28,436
jump discontinuity we call it. We'll talk later, we actually say it's continuous

291
00:24:28,436 --> 00:24:30,060
from the left

292
00:24:30,060 --> 00:24:34,700
because the left-hand limit equals the function value.

293
00:24:35,520 --> 00:24:41,587
It's discontinuous from the right because the limit from the right does not equal

294
00:24:41,587 --> 00:24:42,337
the

295
00:24:42,020 --> 00:24:42,920
function value.

296
00:24:43,140 --> 00:24:45,920
But overall it's definitely discontinuous at zero.

297
00:24:46,080 --> 00:24:51,400
What is the limit as x approaches 2 from the left?

298
00:24:56,160 --> 00:24:57,060
2.

299
00:24:58,440 --> 00:25:01,620
The limit as x approaches 2 from the right?

300
00:25:03,420 --> 00:25:04,320
0.

301
00:25:05,260 --> 00:25:15,660
The two-sided limit as x approaches 2 does not exist because the left-hand

302
00:25:15,660 --> 00:25:24,020
limit does not equal the right-hand limit. And f(2) = 1.

303
00:25:25,980 --> 00:25:32,760
It's definitely discontinuous at 2. Is it continuous on the left? No because the

304
00:25:32,760 --> 00:25:35,800
left-hand limit does not equal the function value.

305
00:25:36,080 --> 00:25:37,080
Is it continuous from the right?

306
00:25:37,660 --> 00:25:40,460
No, because the right-hand limit,

307
00:25:40,620 --> 00:25:42,820
zero does not equal to the function value.

308
00:25:43,900 --> 00:25:44,800
Okay.

309
00:25:44,280 --> 00:25:46,600
Unlike at x = 0, it is not continuous from the left.

310
00:26:00,580 --> 00:26:05,760
What is the limit as x approaches 4 from the left?

311
00:26:10,880 --> 00:26:11,960
This gets closer to 3.

312
00:26:13,180 --> 00:26:16,080
3, limit as x approaches 4 from the right.

313
00:26:17,340 --> 00:26:18,540
3, 3.

314
00:26:19,640 --> 00:26:21,680
What is the limit as x approaches 4?

315
00:26:22,300 --> 00:26:23,200
3.

316
00:26:24,840 --> 00:26:26,640
And what is f of 4?

317
00:26:26,960 --> 00:26:27,860
3.

318
00:26:27,900 --> 00:26:28,800
3.

319
00:26:29,300 --> 00:26:30,200
Okay.

320
00:26:31,080 --> 00:26:33,240
Is the function continuous at 4?

321
00:26:33,720 --> 00:26:34,620
Yes.

322
00:26:34,400 --> 00:26:35,300
Yes.

323
00:26:34,700 --> 00:26:41,420
If we weren't given a graph, if this limit equals the function value, it's

324
00:26:41,420 --> 00:26:42,170
continuous.

325
00:26:43,640 --> 00:26:45,240
Okay, and we're going to define later.

326
00:26:45,340 --> 00:26:49,520
So that means it's continuous at a point, but later it's continuous on an interval.

327
00:26:51,400 --> 00:26:53,540
Okay, I'm hoping you're getting the point.

328
00:26:54,000 --> 00:26:56,900
Let's do, I think I got one more for you.

329
00:26:57,160 --> 00:26:59,080
Anybody know what function this is?

330
00:27:00,760 --> 00:27:01,660
This side.

331
00:27:01,740 --> 00:27:04,140
This is tangent.

332
00:27:09,280 --> 00:27:14,780
What is the limit as x approaches pi from the left?

333
00:27:17,920 --> 00:27:18,820
Zero.

334
00:27:19,960 --> 00:27:22,680
What is the limit as x approaches pi from the right?

335
00:27:23,320 --> 00:27:24,220
Zero.

336
00:27:26,360 --> 00:27:28,860
What is the limit as x approaches pi?

337
00:27:29,720 --> 00:27:30,620
Zero.

338
00:27:30,320 --> 00:27:31,220
Zero.

339
00:27:31,160 --> 00:27:37,380
As x approaches pi over 2 from the left, the tangent values grow larger and larger.

340
00:27:37,380 --> 00:27:38,280
and larger.

341
00:27:39,120 --> 00:27:41,040
So we put positive infinity.

342
00:27:48,300 --> 00:27:52,720
Now consider the limit as x approaches pi over 2 from the right.

343
00:27:57,060 --> 00:28:02,300
It approaches negative infinity; from the right, the graph keeps going down

344
00:28:02,300 --> 00:28:13,280
forever. The two-sided limit as x approaches pi over 2 does not exist

345
00:28:13,280 --> 00:28:19,260
because the one-sided limits are not equal. If both approached positive infinity, we

346
00:28:19,260 --> 00:28:20,180
would write

347
00:28:20,180 --> 00:28:21,960
positive in here, they both go down.

348
00:28:22,780 --> 00:28:23,880
But they go in different direction,

349
00:28:24,220 --> 00:28:25,120
definitely doesn't exist.

350
00:28:32,560 --> 00:28:34,800
So is it continuous at pi over two?

351
00:28:37,980 --> 00:28:41,560
Definitely not, in fact, what is tan of pi over two?

352
00:28:44,960 --> 00:28:45,960
Does not exist.

353
00:28:47,940 --> 00:28:48,840
Okay.

354
00:28:50,180 --> 00:28:53,220
What do we call this type of discontinuity?

355
00:28:56,440 --> 00:28:59,540
We just call it infinite discontinuity.

356
00:29:01,100 --> 00:29:05,240
A hole in the graph we call a removable discontinuity.

357
00:29:07,160 --> 00:29:09,200
If it jumps, it's called a jump discontinuity.

358
00:29:09,260 --> 00:29:10,180
OK.

359
00:29:11,360 --> 00:29:14,940
This is a function of sine of pi over x.

360
00:29:21,360 --> 00:29:22,260
OK.

361
00:29:21,600 --> 00:29:24,500
It's continuous everywhere except to what?

362
00:29:26,080 --> 00:29:31,480
At x = 0 the function is undefined, but for every other x we can evaluate sine of

363
00:29:31,480 --> 00:29:32,230
that value.

364
00:29:33,500 --> 00:29:36,140
The interesting part the limit has

365
00:29:37,620 --> 00:29:39,740
X approaches zero

366
00:29:44,260 --> 00:29:48,180
Okay to get closer to closer to zero what's happening

367
00:29:51,920 --> 00:29:54,260
So let's look at this as x goes to zero,

368
00:29:55,320 --> 00:29:57,140
what's happening to this?

369
00:29:59,520 --> 00:30:02,000
What's pi over a very, very small number?

370
00:30:04,520 --> 00:30:09,300
Like pi divided by 0.00001 is a very big number.

371
00:30:10,680 --> 00:30:13,780
The closer x gets to zero, the larger pi divided by x becomes in magnitude.

372
00:30:15,680 --> 00:30:18,100
And what is sine of a very, very big number?

373
00:30:21,120 --> 00:30:23,040
It just keeps doing this forever.

374
00:30:23,240 --> 00:30:25,560
It keeps oscillating.

375
00:30:26,400 --> 00:30:31,360
As x approaches zero, the inside approaches infinity in magnitude, and the

376
00:30:31,360 --> 00:30:33,180
inside just keeps oscillating back and forth.

377
00:30:34,000 --> 00:30:34,980
You get this really cool path.

378
00:30:35,440 --> 00:30:41,760
So this, both from the left and from the right, don't approach a single number.

379
00:30:42,600 --> 00:30:46,320
It never exceeds 1 or goes below negative 1, but

380
00:30:46,320 --> 00:30:49,020
It does not approach a number, so this does not exist.

381
00:30:51,020 --> 00:30:53,540
It's called an oscillating discontinuity.

382
00:30:54,480 --> 00:30:57,180
It's actually really cool if you look at the on Desmos.

383
00:31:20,720 --> 00:31:24,500
The more I zoom in, the more there are.

384
00:31:27,520 --> 00:31:29,380
It completely fills up the whole screen.

385
00:31:30,540 --> 00:31:33,780
In reality, there is a tiny gap between every one of those lines.

386
00:31:35,080 --> 00:31:38,080
But the closer you get, the faster it goes back and forth, back and forth.

387
00:31:39,280 --> 00:31:40,180
Super cool.

388
00:31:42,320 --> 00:31:45,200
Okay, very weird as well.

389
00:31:46,180 --> 00:31:50,380
But there we go.

390
00:31:50,780 --> 00:31:53,500
Okay I think we're going to stop here.

391
00:31:57,620 --> 00:32:02,709
Again all the limits, the idea of left-hand limit, right-hand limit, the limits,

392
00:32:02,709 --> 00:32:03,459
it's

393
00:32:03,210 --> 00:32:08,390
only based on tables of values and graphs and Friday we'll talk about when

394
00:32:08,390 --> 00:32:10,430
we are given the actual equation.

395
00:32:11,250 --> 00:32:15,690
We're just going to, I put a new Delta Math assignment up, so just do Delta Math.

396
00:32:16,930 --> 00:32:17,830
So you can get going on that.