1
00:00:00,000 --> 00:00:03,800
- [Instructor] In a previous
video, we explored the graphs

2
00:00:00,000 --> 00:00:00,880
of Y equals one over X
squared and one over X.

3
00:00:00,880 --> 00:00:02,980
In a previous video we've
looked at these graphs.

4
00:00:02,980 --> 00:00:05,350
This is Y is equal to one over X squared.

5
00:00:05,350 --> 00:00:07,840
This is Y is equal to one over X.

6
00:00:07,840 --> 00:00:09,700
And we explored what's the limit

7
00:00:09,700 --> 00:00:13,850
as X approaches zero in
either of those scenarios.

8
00:00:13,850 --> 00:00:15,820
And in this left scenario we saw

9
00:00:15,820 --> 00:00:18,340
as X becomes less and less negative,

10
00:00:18,340 --> 00:00:22,916
as it approaches zero
from the left hand side,

11
00:00:22,916 --> 00:00:26,180
the value of one over
X squared is unbounded

12
00:00:26,180 --> 00:00:27,560
in the positive direction.

13
00:00:27,560 --> 00:00:30,930
And the same thing happens as
we approach X from the right,

14
00:00:30,930 --> 00:00:32,530
as we become less and less positive

15
00:00:32,530 --> 00:00:34,170
but we are still positive,

16
00:00:34,170 --> 00:00:35,990
the value of one over X squared becomes

17
00:00:35,990 --> 00:00:38,010
unbounded in the positive direction.

18
00:00:38,010 --> 00:00:39,727
So in that video, we just said, "Hey,

19
00:00:39,727 --> 00:00:43,150
"one could say that this
limit is unbounded."

20
00:00:43,150 --> 00:00:45,270
But what we're going
to do in this video is

21
00:00:45,270 --> 00:00:47,410
introduce new notation.

22
00:00:47,410 --> 00:00:49,150
Instead of just saying it's unbounded,

23
00:00:49,150 --> 00:00:51,160
we could say, "Hey, from
both the left and the right

24
00:00:51,160 --> 00:00:53,590
it looks like we're going
to positive infinity".

25
00:00:53,590 --> 00:00:55,697
So we can introduce
this notation of saying,

26
00:00:55,697 --> 00:00:58,320
"Hey, this is going to infinity",

27
00:00:58,320 --> 00:01:00,240
which you will sometimes see used.

28
00:01:00,240 --> 00:01:01,700
Some people would call this unbounded,

29
00:01:01,700 --> 00:01:03,160
some people say it does not exist

30
00:01:03,160 --> 00:01:05,730
because it's not approaching
some finite value,

31
00:01:05,730 --> 00:01:07,630
while some people will use this notation

32
00:01:07,630 --> 00:01:10,220
of the limit going to infinity.

33
00:01:10,220 --> 00:01:11,760
But what about this scenario?

34
00:01:11,760 --> 00:01:14,220
Can we use our new notation here?

35
00:01:14,220 --> 00:01:18,310
Well, when we approach zero from the left,

36
00:01:18,310 --> 00:01:21,120
it looks like we're unbounded
in the negative direction,

37
00:01:21,120 --> 00:01:23,310
and when we approach zero from the right,

38
00:01:23,310 --> 00:01:26,260
we are unbounded in
the positive direction.

39
00:01:26,260 --> 00:01:28,760
So, here you still could not say

40
00:01:28,760 --> 00:01:30,710
that the limit is approaching infinity

41
00:01:30,710 --> 00:01:32,440
because from the right
it's approaching infinity,

42
00:01:32,440 --> 00:01:34,660
but from the left it's
approaching negative infinity.

43
00:01:34,660 --> 00:01:39,660
So you would still say
that this does not exist.

44
00:01:39,760 --> 00:01:42,340
You could do one sided limits here,

45
00:01:42,340 --> 00:01:43,670
which if you're not familiar with,

46
00:01:43,670 --> 00:01:45,780
I encourage you to review
it on Khan Academy.

47
00:01:45,780 --> 00:01:49,010
If you said the limit of one over X

48
00:01:49,010 --> 00:01:53,340
as X approaches zero
from the left hand side,

49
00:01:53,340 --> 00:01:55,810
from values less than zero,

50
00:01:55,810 --> 00:01:57,627
well then you would look at
this right over here and say,

51
00:01:57,627 --> 00:01:59,544
"Well, look, it looks like we're going

52
00:01:59,544 --> 00:02:00,790
unbounded in the negative direction".

53
00:02:00,790 --> 00:02:04,270
So you would say this is
equal to negative infinity.

54
00:02:04,270 --> 00:02:09,270
And of course if you said the
limit as X approaches zero

55
00:02:09,669 --> 00:02:12,700
from the right of one over X, well here

56
00:02:12,700 --> 00:02:14,500
you're unbounded in the positive direction

57
00:02:14,500 --> 00:02:17,650
so that's going to be
equal to positive infinity.

58
00:02:17,650 --> 00:02:19,760
Let's do an example
problem from Khan Academy

59
00:02:19,760 --> 00:02:22,493
based on this idea and this notation.

60
00:02:23,610 --> 00:02:27,540
So here it says, consider
graphs A, B, and C.

61
00:02:27,540 --> 00:02:30,470
The dashed lines represent asymptotes.

62
00:02:30,470 --> 00:02:33,260
Which of the graphs agree
with this statement,

63
00:02:33,260 --> 00:02:36,160
that the limit as X approaches 1 of H of X

64
00:02:36,160 --> 00:02:37,480
is equal to infinity?

65
00:02:37,480 --> 00:02:39,980
Pause this video and see
if you can figure it out.

66
00:02:40,940 --> 00:02:42,350
Alright, let's go through each of these.

67
00:02:42,350 --> 00:02:44,850
So we want to think about
what happens at X equals one.

68
00:02:44,850 --> 00:02:47,860
So that's right over here on graph A.

69
00:02:47,860 --> 00:02:49,880
So as we approach X equals one,

70
00:02:49,880 --> 00:02:52,120
so let me write this, so the limit,

71
00:02:52,120 --> 00:02:53,860
let me do this for the different graphs.

72
00:02:53,860 --> 00:02:58,753
So, for graph A, the
limit as x approaches one

73
00:02:59,680 --> 00:03:02,360
from the left, that looks like

74
00:03:02,360 --> 00:03:04,160
it's unbounded in the positive direction.

75
00:03:04,160 --> 00:03:07,091
That equals infinity and the limit

76
00:03:07,091 --> 00:03:11,530
as X approaches one from the right,

77
00:03:11,530 --> 00:03:14,020
well that looks like it's
going to negative infinity.

78
00:03:14,020 --> 00:03:15,970
That equals negative infinity.

79
00:03:15,970 --> 00:03:18,770
And since these are going
in two different directions,

80
00:03:18,770 --> 00:03:19,860
you wouldn't be able to say that

81
00:03:19,860 --> 00:03:21,420
the limit as X approaches one

82
00:03:21,420 --> 00:03:23,450
from both directions is equal to infinity.

83
00:03:23,450 --> 00:03:25,700
So I would rule this one out.

84
00:03:25,700 --> 00:03:27,710
Now let's look at choice B.

85
00:03:27,710 --> 00:03:32,710
What's the limit as X
approaches one from the left?

86
00:03:33,220 --> 00:03:36,250
And of course these are of H of X.

87
00:03:36,250 --> 00:03:37,610
Gotta write that down.

88
00:03:37,610 --> 00:03:40,970
So, of H of X right over here.

89
00:03:40,970 --> 00:03:43,589
Well, as we approach from the left,

90
00:03:43,589 --> 00:03:47,390
looks like we're going
to positive infinity.

91
00:03:47,390 --> 00:03:50,740
And it looks like the limit of H of X

92
00:03:50,740 --> 00:03:54,220
as we approach one from the right is

93
00:03:54,220 --> 00:03:56,860
also going to positive infinity.

94
00:03:56,860 --> 00:03:58,710
And so, since we're
approaching you could say

95
00:03:58,710 --> 00:04:02,630
the same direction of infinity,
you could say this for B.

96
00:04:02,630 --> 00:04:04,490
So B meets the constraints, but

97
00:04:04,490 --> 00:04:06,730
let's just check C to make sure.

98
00:04:06,730 --> 00:04:09,890
Well, you can see very
clearly X equals one,

99
00:04:09,890 --> 00:04:11,230
that as we approach it from the left,

100
00:04:11,230 --> 00:04:12,490
we go to negative infinity,

101
00:04:12,490 --> 00:04:14,712
and as we approach from the right,

102
00:04:14,712 --> 00:04:15,545
we got to positive infinity.

103
00:04:16,401 --> 00:04:18,740
So this, once again,
would not be approaching

104
00:04:18,740 --> 00:04:19,880
the same infinity.

105
00:04:19,880 --> 00:04:22,293
So you would rule this one out, as well.