1
00:00:01,430 --> 00:00:05,265
In this video, we'll define what
it means for a function F to

2
00:00:05,265 --> 00:00:07,625
tend to a limit as X tends to

3
00:00:07,625 --> 00:00:12,085
Infinity. We also define what it
means for its tend to limit as X

4
00:00:12,085 --> 00:00:15,025
tends to minus Infinity, and as
X tends to a real number.

5
00:00:17,440 --> 00:00:21,059
Remember from the video limits
of sequences, we define what it

6
00:00:21,059 --> 00:00:25,994
meant for a sequence to tend to
A to a limit as N tends to

7
00:00:25,994 --> 00:00:32,274
Infinity, we said. Sequence
wired tends to limit L

8
00:00:32,274 --> 00:00:36,169
as an tends to Infinity.

9
00:00:37,190 --> 00:00:40,112
If however, small a number I

10
00:00:40,112 --> 00:00:46,504
chose. The sequence YN would get
that close to L and stay that

11
00:00:46,504 --> 00:00:51,957
close. One of our examples
was the sequence YN equals

12
00:00:51,957 --> 00:00:53,346
one over N.

13
00:00:54,480 --> 00:00:56,916
I just put a graph of this.

14
00:01:01,720 --> 00:01:05,440
So her plot the points
of why and which was

15
00:01:05,440 --> 00:01:07,300
one equals one over N.

16
00:01:08,400 --> 00:01:11,920
It looks like this sort of it 1.

17
00:01:12,500 --> 00:01:16,236
And then two was a half. Then it

18
00:01:16,236 --> 00:01:23,155
went 1/3. Now this
sequence tended to

19
00:01:23,155 --> 00:01:26,380
0 because eventually.

20
00:01:26,930 --> 00:01:29,390
The points in the sequence get

21
00:01:29,390 --> 00:01:34,844
as close. Zeros alike, so have a
smaller distance I choose.

22
00:01:38,010 --> 00:01:41,475
The points in the sequence
will eventually get within

23
00:01:41,475 --> 00:01:44,555
that boundary from zero and
stay in there.

24
00:01:45,710 --> 00:01:48,900
The functions are defined
in a similar way to limits

25
00:01:48,900 --> 00:01:49,538
of sequences.

26
00:01:50,950 --> 00:01:54,184
If we have a function like FX
equals one over X.

27
00:01:54,740 --> 00:01:57,310
Where I've actually already
lost some of the points for

28
00:01:57,310 --> 00:02:00,137
this already, so I'll use
it if we sketch the graph.

29
00:02:02,410 --> 00:02:06,994
You can see that this function
gets closer and closer to 0.

30
00:02:07,690 --> 00:02:09,410
In fact, whichever tiny distance

31
00:02:09,410 --> 00:02:13,390
we choose. Like this distance
I've already sketched in here.

32
00:02:14,300 --> 00:02:18,330
Whatever distance we choose this
function will get within that

33
00:02:18,330 --> 00:02:20,748
from zero and stay that close.

34
00:02:21,650 --> 00:02:25,544
If a function does this,
we say it tends to limit

35
00:02:25,544 --> 00:02:26,960
O's extends to Infinity.

36
00:02:28,070 --> 00:02:32,390
If a function tends to zeros
extends to Infinity, we write it

37
00:02:32,390 --> 00:02:38,426
like this. Who writes FX
tends to 0?

38
00:02:39,330 --> 00:02:41,120
As X tends to Infinity.

39
00:02:42,890 --> 00:02:50,157
For sometimes the limits of FX
as X tends to Infinity equals 0.

40
00:02:52,290 --> 00:02:58,442
Here's another
function with

41
00:02:58,442 --> 00:03:01,518
the limit.

42
00:03:03,000 --> 00:03:05,324
This time will have F of X.

43
00:03:05,970 --> 00:03:12,009
Equals 3 - 1 over X squared,
4X greater than 0.

44
00:03:13,580 --> 00:03:17,216
So I'll just sketch this graph.

45
00:03:18,410 --> 00:03:20,180
Will have three about here.

46
00:03:29,040 --> 00:03:31,350
Right?

47
00:03:32,720 --> 00:03:34,820
This graph looks
something like this.

48
00:03:42,650 --> 00:03:45,836
Now again, whatever tiny
distance I choose around 3:00.

49
00:03:48,450 --> 00:03:50,389
It doesn't matter
how small this is.

50
00:03:51,990 --> 00:03:53,060
This function.

51
00:03:55,490 --> 00:03:59,050
Eventually gets trapped that far
away from 3.

52
00:03:59,890 --> 00:04:05,467
So this function has limit
three, and again we write FX

53
00:04:05,467 --> 00:04:06,988
tends to three.

54
00:04:08,050 --> 00:04:09,620
As X tends to Infinity.

55
00:04:10,210 --> 00:04:13,670
For the limit of FX.

56
00:04:13,670 --> 00:04:16,659
Equals 3 as X tends to Infinity.

57
00:04:18,800 --> 00:04:23,389
In general, we say that the
function has limit L as X tends

58
00:04:23,389 --> 00:04:26,566
to Infinity. If how the smaller
distance he choose.

59
00:04:27,190 --> 00:04:31,282
The function gets closer than
that and stays closer than that

60
00:04:31,282 --> 00:04:32,770
as X gets larger.

61
00:04:35,030 --> 00:04:42,095
Not all functions have
real limits as X tends

62
00:04:42,095 --> 00:04:46,805
to Infinity. Here's one
that doesn't.

63
00:04:48,100 --> 00:04:52,042
Will have F

64
00:04:52,042 --> 00:04:55,768
of X. Equals X squared.

65
00:04:56,660 --> 00:04:58,990
And that is, I'm sure, you
know, looks something like

66
00:04:58,990 --> 00:04:59,223
this.

67
00:05:05,500 --> 00:05:08,954
Now as X gets larger, this
certainly doesn't get any closer

68
00:05:08,954 --> 00:05:10,210
to any real number.

69
00:05:11,380 --> 00:05:14,208
In fact, however, large
number I choose.

70
00:05:16,480 --> 00:05:20,300
This function will eventually
get bigger and stay bigger than

71
00:05:20,300 --> 00:05:22,974
it as X goes off to Infinity.

72
00:05:24,230 --> 00:05:28,808
If a function does this, we say
that it has limits Infinity as X

73
00:05:28,808 --> 00:05:35,384
tends to Infinity. We write F
of X tends to Infinity.

74
00:05:36,290 --> 00:05:37,980
As X tends to Infinity.

75
00:05:38,490 --> 00:05:45,096
Or limits. With X as
X tends to Infinity equals

76
00:05:45,096 --> 00:05:52,178
Infinity. If we
have a function like

77
00:05:52,178 --> 00:05:55,950
F of X equals

78
00:05:55,950 --> 00:06:02,038
minus X. It
looks like this.

79
00:06:04,300 --> 00:06:11,860
This also doesn't tend to a
real limit as X tends to

80
00:06:11,860 --> 00:06:15,707
Infinity. But it doesn't tend to
Infinity either, because it

81
00:06:15,707 --> 00:06:16,799
doesn't get very large.

82
00:06:17,510 --> 00:06:19,376
In fact, have a large and

83
00:06:19,376 --> 00:06:21,458
negative. A number I choose.

84
00:06:22,670 --> 00:06:26,311
This function will get below
that number and stay below it.

85
00:06:27,230 --> 00:06:30,674
If a function does this, we'd
say it tends to minus Infinity,

86
00:06:30,674 --> 00:06:32,109
as X tends to Infinity.

87
00:06:33,190 --> 00:06:36,320
So we write. F of X.

88
00:06:36,870 --> 00:06:38,558
Tends to minus Infinity.

89
00:06:39,360 --> 00:06:41,100
As X tends to Infinity.

90
00:06:41,830 --> 00:06:45,148
Oregon limit of F of X.

91
00:06:45,980 --> 00:06:49,756
Equals minus Infinity
as X tends to Infinity.

92
00:06:50,910 --> 00:06:56,038
Some functions that
have any limits at all,

93
00:06:56,038 --> 00:06:59,243
as X tends to Infinity.

94
00:07:00,330 --> 00:07:06,765
This is the graph FX
equals X sign X.

95
00:07:08,170 --> 00:07:11,368
But as you can see, it certainly
doesn't get closer to any real

96
00:07:11,368 --> 00:07:13,588
number. As extends to Infinity.

97
00:07:15,090 --> 00:07:17,866
Also, if I pick a really
large number.

98
00:07:21,310 --> 00:07:22,835
The graph will eventually get

99
00:07:22,835 --> 00:07:26,004
above it. But it won't stay
above it because it always

100
00:07:26,004 --> 00:07:27,732
come back down to zero and
go negative.

101
00:07:28,770 --> 00:07:31,630
Also, for pick a really large
negative number, it will go

102
00:07:31,630 --> 00:07:34,490
below it, but it won't stay
below it because it'll go

103
00:07:34,490 --> 00:07:36,830
back up to zero and then go
positive again.

104
00:07:37,990 --> 00:07:41,782
So this function doesn't tend to
Infinity, and it doesn't tend to

105
00:07:41,782 --> 00:07:43,362
in minus Infinity as well.

106
00:07:44,290 --> 00:07:46,458
So this function doesn't
have any limited tool.

107
00:07:47,800 --> 00:07:54,598
We can also define limits
for functions as X tends to

108
00:07:54,598 --> 00:07:55,834
minus Infinity.

109
00:07:57,460 --> 00:08:01,140
Here's a graph of E to the X.

110
00:08:03,160 --> 00:08:06,652
So. This after 0 gets

111
00:08:06,652 --> 00:08:09,170
large. The 40

112
00:08:10,410 --> 00:08:12,486
Gets closer and closer to 0.

113
00:08:15,410 --> 00:08:19,388
As X gets more negative here.

114
00:08:19,940 --> 00:08:22,940
This function gets closer and
closer to 0.

115
00:08:24,320 --> 00:08:27,449
And however small a
distance I choose.

116
00:08:29,320 --> 00:08:35,282
The function will get closer
than that to zero and stay

117
00:08:35,282 --> 00:08:39,076
closer than that as X gets more

118
00:08:39,076 --> 00:08:42,720
negative. If a function is this

119
00:08:42,720 --> 00:08:46,614
we say? F of X tends

120
00:08:46,614 --> 00:08:52,796
to. 0. As
X tends to minus Infinity.

121
00:08:53,850 --> 00:09:00,174
Oh, limit as X tends to minus
Infinity of F of X.

122
00:09:00,860 --> 00:09:01,850
Equals 0.

123
00:09:03,590 --> 00:09:08,678
In general, we say a function
has limit L's extends to minus

124
00:09:08,678 --> 00:09:12,797
Infinity. If how the smaller
distance I choose the function

125
00:09:12,797 --> 00:09:17,256
gets closer than that and stays
closer than that as X gets more

126
00:09:17,256 --> 00:09:22,745
negative. If we
have a graph

127
00:09:22,745 --> 00:09:25,430
like X squared.

128
00:09:26,620 --> 00:09:28,560
Shoulder sketch.

129
00:09:34,390 --> 00:09:41,254
You can see that this doesn't
get closer to any real number as

130
00:09:41,254 --> 00:09:43,894
X goes to minus Infinity.

131
00:09:44,970 --> 00:09:47,298
But however larger
number I choose.

132
00:09:50,030 --> 00:09:51,760
As X gets more negative.

133
00:09:52,340 --> 00:09:55,580
This graph will go above
that number and stay above

134
00:09:55,580 --> 00:09:56,228
that number.

135
00:09:57,660 --> 00:10:02,070
If a function does this, we say
it tends to Infinity as X tends

136
00:10:02,070 --> 00:10:08,074
to minus Infinity. So F of
X tends to Infinity.

137
00:10:08,940 --> 00:10:12,816
As X tends to minus Infinity.

138
00:10:12,820 --> 00:10:19,099
Limits of F of X equals Infinity
as X tends to minus Infinity.

139
00:10:19,870 --> 00:10:26,886
Now function like X
cubed doesn't do either

140
00:10:26,886 --> 00:10:33,902
of these things. The
graph is something like

141
00:10:33,902 --> 00:10:34,779
this.

142
00:10:36,340 --> 00:10:39,520
So it certainly doesn't get
larger than any number I pick as

143
00:10:39,520 --> 00:10:42,170
X goes to minus Infinity and
stay larger than it.

144
00:10:42,710 --> 00:10:45,544
And it doesn't get closer to any
real number, so it doesn't have

145
00:10:45,544 --> 00:10:50,225
a real limit. But however larger
negative number I choose.

146
00:10:52,240 --> 00:10:55,595
As X gets more negative, this
function will drop below that

147
00:10:55,595 --> 00:10:59,560
number and stay below it and
this works for any number I can

148
00:10:59,560 --> 00:11:06,010
choose. When this happens, we
say F of X tends to minus

149
00:11:06,010 --> 00:11:09,080
Infinity. As X tends to minus

150
00:11:09,080 --> 00:11:13,270
Infinity. For, again, the limits
of F of X.

151
00:11:13,980 --> 00:11:15,609
Equals minus Infinity.

152
00:11:16,140 --> 00:11:18,120
As X tends to minus Infinity.

153
00:11:20,130 --> 00:11:25,602
Now again, a function didn't
have any limit as X tends to

154
00:11:25,602 --> 00:11:29,836
minus Infinity. And our graph
of FX equals X sign X is a

155
00:11:29,836 --> 00:11:31,016
good example of this again.

156
00:11:32,250 --> 00:11:35,341
As X gets more negative, this
certainly doesn't get closer to

157
00:11:35,341 --> 00:11:38,151
any real number, so it doesn't
have a real limit.

158
00:11:38,710 --> 00:11:41,817
It might go above any number we
can pick, but it won't stay

159
00:11:41,817 --> 00:11:44,446
above it because it comes back
down and goes through zero.

160
00:11:45,130 --> 00:11:48,510
And it won't get more. It won't
get more and stay more negative

161
00:11:48,510 --> 00:11:51,110
than any large negative number
we can choose, because always

162
00:11:51,110 --> 00:11:52,930
come back up to zero and go

163
00:11:52,930 --> 00:11:57,151
positive again. So this is a
function that has no limit as X

164
00:11:57,151 --> 00:11:58,195
tends to minus Infinity.

165
00:11:59,540 --> 00:12:06,028
There's one more type
of limit. We can

166
00:12:06,028 --> 00:12:10,407
define functions. Let's
look at the graph of the

167
00:12:10,407 --> 00:12:13,417
really simple function
like F of X equals X +3.

168
00:12:15,870 --> 00:12:18,960
So that looks something like

169
00:12:18,960 --> 00:12:20,390
this. Three there.

170
00:12:21,760 --> 00:12:28,456
If I pick a
point on this graph,

171
00:12:28,456 --> 00:12:30,130
like say.

172
00:12:31,590 --> 00:12:33,000
Here, when X is one.

173
00:12:35,740 --> 00:12:37,498
The graph goes through 1 four.

174
00:12:39,810 --> 00:12:42,337
You can see that as X approaches

175
00:12:42,337 --> 00:12:45,998
one. The value of the
function approaches 4.

176
00:12:47,850 --> 00:12:50,310
We say that.

177
00:12:50,310 --> 00:12:52,830
F of X the limit of F of X.

178
00:12:53,480 --> 00:12:56,777
Equals 4 as X tends to one.

179
00:12:58,370 --> 00:12:59,230
Similarly.

180
00:13:00,470 --> 00:13:02,960
If we say X equals 5.

181
00:13:04,370 --> 00:13:06,209
Then the function.

182
00:13:07,550 --> 00:13:12,647
Is it 8? And his ex gets closer
to five. The function gets close

183
00:13:12,647 --> 00:13:19,612
to 8. So the limit of F
of X as X tends to 5

184
00:13:19,612 --> 00:13:20,564
equals 8.

185
00:13:22,550 --> 00:13:24,000
Now this doesn't look very

186
00:13:24,000 --> 00:13:27,700
useful. But it does become
useful if we have functions that

187
00:13:27,700 --> 00:13:28,856
aren't defined data point.

188
00:13:29,880 --> 00:13:36,640
This is the case with
this function. This is F

189
00:13:36,640 --> 00:13:39,690
of X. Equals.

190
00:13:40,230 --> 00:13:43,330
X sign one over X.

191
00:13:45,200 --> 00:13:48,905
You can see for this function
that as X gets close to 0.

192
00:13:49,680 --> 00:13:50,960
The function gets very very

193
00:13:50,960 --> 00:13:55,980
close to 0. But the cause at X
equals 0. We have a zero

194
00:13:55,980 --> 00:13:59,644
denominator. This function isn't
defined for X equals 0.

195
00:14:01,510 --> 00:14:04,821
But because the function gets
closer and closer to zero, it's

196
00:14:04,821 --> 00:14:08,734
almost as if the value of the
function at zero should be 0.

197
00:14:10,780 --> 00:14:14,434
So the limit of F of X.

198
00:14:15,330 --> 00:14:18,760
As X tends to 0 equals 0.

199
00:14:19,330 --> 00:14:20,818
And you can think of 0.

200
00:14:21,340 --> 00:14:25,292
As the value F of X should
take when X equals 0, even

201
00:14:25,292 --> 00:14:26,508
though it's not defined.

202
00:14:28,030 --> 00:14:30,958
Here's another example.

203
00:14:30,960 --> 00:14:37,552
This function is F
of X equals Y

204
00:14:37,552 --> 00:14:43,320
to the minus one
over X squared.

205
00:14:44,520 --> 00:14:48,576
Now again, when X equals 0, we
have a zero denominator. Here in

206
00:14:48,576 --> 00:14:52,098
the function. So it's not
defined at X equals 0.

207
00:14:53,340 --> 00:14:56,308
But as X gets close to 0.

208
00:14:56,960 --> 00:15:02,462
This function approaches 0, so
again the limit of F of X as X

209
00:15:02,462 --> 00:15:07,964
tends to zero in this case is
zero, and again we can think of

210
00:15:07,964 --> 00:15:12,287
0. Is the value F should take
when X equals 0.

211
00:15:14,360 --> 00:15:21,512
Not all functions
have nice limits

212
00:15:21,512 --> 00:15:23,896
like this.

213
00:15:25,090 --> 00:15:31,966
If we look at the function
MoD X over XF of X

214
00:15:31,966 --> 00:15:34,831
equals MoD X over X.

215
00:15:36,510 --> 00:15:38,070
If X is positive.

216
00:15:39,020 --> 00:15:44,578
Then The
top of this

217
00:15:44,578 --> 00:15:46,879
is just X.

218
00:15:48,090 --> 00:15:51,879
So this becomes X over X,
which is one.

219
00:15:52,960 --> 00:15:55,214
But if X is less than 0.

220
00:15:55,840 --> 00:16:01,043
This becomes little becomes
minus X, so that's minus X over

221
00:16:01,043 --> 00:16:03,510
X. So that's minus one.

222
00:16:04,550 --> 00:16:08,694
When X is 0, is not defined at
all, so the graph looks like

223
00:16:08,694 --> 00:16:10,898
this. It's just one.

224
00:16:12,480 --> 00:16:14,566
When X is greater than zero and

225
00:16:14,566 --> 00:16:16,900
it's just. Minus one.

226
00:16:19,650 --> 00:16:21,006
When X is less than 0.

227
00:16:22,590 --> 00:16:26,300
And you can see here there's no
way we can define a good limit

228
00:16:26,300 --> 00:16:29,480
for this, because as we approach
zero from the right hand side.

229
00:16:30,100 --> 00:16:33,880
It looks like it should be one,
but if we approach zero from the

230
00:16:33,880 --> 00:16:36,850
left hand side, the function
looks like it should be taking

231
00:16:36,850 --> 00:16:41,252
value minus one. So here it's
not clear what ever vex should

232
00:16:41,252 --> 00:16:42,842
be when X is 0.

233
00:16:43,390 --> 00:16:46,349
Because there isn't a proper
limit for this function as X

234
00:16:46,349 --> 00:16:49,846
tends to 0, so this is a good
example of a function that

235
00:16:49,846 --> 00:16:52,536
doesn't have a limit as X
tends to a number.