WEBVTT
1
00:00:01.439 --> 00:00:04.570 A:middle L:90%
we want to discuss the SM topic. Behavior of
2
00:00:04.620 --> 00:00:07.040 A:middle L:90%
F F X is equal to X to the fourth
3
00:00:07.049 --> 00:00:09.289 A:middle L:90%
plus one over X, the same manner that we
4
00:00:09.289 --> 00:00:12.160 A:middle L:90%
did in exercise 74. And then we want to
5
00:00:12.160 --> 00:00:15.269 A:middle L:90%
use this to help us sketch the breath. So
6
00:00:15.269 --> 00:00:20.269 A:middle L:90%
in 74 they give us what they believe the psychotic
7
00:00:20.280 --> 00:00:23.480 A:middle L:90%
behavior should be. Ah, and this one we
8
00:00:23.480 --> 00:00:25.980 A:middle L:90%
need to figure out what we think it might be
9
00:00:27.390 --> 00:00:34.750 A:middle L:90%
. So if we were to first divide X into
10
00:00:35.060 --> 00:00:38.390 A:middle L:90%
exile before and one, we would end up with
11
00:00:38.939 --> 00:00:42.729 A:middle L:90%
X cubed plus one over X. Now, if
12
00:00:42.729 --> 00:00:46.369 A:middle L:90%
you look at this well, we know that X
13
00:00:46.369 --> 00:00:50.289 A:middle L:90%
cubed is going to tend to plus or minus infinity
14
00:00:50.500 --> 00:00:52.549 A:middle L:90%
, and we know that one over X is going
15
00:00:52.549 --> 00:00:55.439 A:middle L:90%
to tend to zero as X goes to plus or
16
00:00:55.439 --> 00:00:59.000 A:middle L:90%
minus infinity. So as X goes to plus or
17
00:00:59.000 --> 00:01:04.680 A:middle L:90%
minus infinity so essentially, for very large values,
18
00:01:06.120 --> 00:01:08.310 A:middle L:90%
this one over X doesn't really matter. And this
19
00:01:08.319 --> 00:01:11.890 A:middle L:90%
X cubed is a thing that truly matters. So
20
00:01:11.890 --> 00:01:19.459 A:middle L:90%
what we want to show is that the limit as
21
00:01:19.549 --> 00:01:25.750 A:middle L:90%
X approaches, plus or minus infinity of X cubed
22
00:01:26.939 --> 00:01:30.370 A:middle L:90%
are not execute but X to the fourth plus one
23
00:01:30.469 --> 00:01:38.849 A:middle L:90%
over X minus X cubed is equal to zero.
24
00:01:38.030 --> 00:01:42.299 A:middle L:90%
So we want to show this because if this is
25
00:01:42.299 --> 00:01:45.599 A:middle L:90%
the case, then we know that the end behavior
26
00:01:45.680 --> 00:01:51.049 A:middle L:90%
for our function f of X really is exe cute
27
00:01:52.799 --> 00:01:55.180 A:middle L:90%
. So let's go ahead and start with that.
28
00:01:55.530 --> 00:01:59.409 A:middle L:90%
So let's just simplify the inside first. So I'll
29
00:01:59.409 --> 00:02:01.709 A:middle L:90%
go ahead and do that up here. So we
30
00:02:01.709 --> 00:02:07.709 A:middle L:90%
have X to the fourth plus one over X minus
31
00:02:07.719 --> 00:02:08.889 A:middle L:90%
x cute. Now we need to get a common
32
00:02:08.889 --> 00:02:12.849 A:middle L:90%
denominator. So let's move by Execute top and bottom
33
00:02:12.849 --> 00:02:16.900 A:middle L:90%
by X simplifying this down. This is going to
34
00:02:16.900 --> 00:02:22.050 A:middle L:90%
give us X to the fourth plus one minus X
35
00:02:22.050 --> 00:02:24.469 A:middle L:90%
to the fourth all over X and well, now
36
00:02:25.129 --> 00:02:30.300 A:middle L:90%
those extra force cancel out and we just be left
37
00:02:30.300 --> 00:02:35.120 A:middle L:90%
with one over X. So all we need to
38
00:02:35.120 --> 00:02:40.650 A:middle L:90%
do is show that now the limit as X approaches
39
00:02:40.650 --> 00:02:45.020 A:middle L:90%
infinity of one of her ex Is he good zero
40
00:02:45.030 --> 00:02:46.659 A:middle L:90%
? Well, that's the century will be said here
41
00:02:46.659 --> 00:02:50.550 A:middle L:90%
at the very start. So that zero and then
42
00:02:50.560 --> 00:02:54.259 A:middle L:90%
likewise as X approaches negative infinity of one of Rex
43
00:02:55.099 --> 00:03:00.169 A:middle L:90%
. Well, that would also be zero. So
44
00:03:00.039 --> 00:03:07.719 A:middle L:90%
we show that this year simplifies down too. One
45
00:03:07.719 --> 00:03:12.509 A:middle L:90%
of these cases and both of the limits end up
46
00:03:12.509 --> 00:03:15.210 A:middle L:90%
giving us zero. So we know the end behavior
47
00:03:15.250 --> 00:03:19.550 A:middle L:90%
of this function. Extra four plus one over X
48
00:03:19.560 --> 00:03:23.159 A:middle L:90%
should be very close to execute. All right,
49
00:03:23.210 --> 00:03:27.210 A:middle L:90%
now that we have that information, let's just go
50
00:03:27.210 --> 00:03:30.500 A:middle L:90%
ahead and find our intercepts at our vertical aspirin tubes
51
00:03:30.610 --> 00:03:37.409 A:middle L:90%
. So we I know that our ex intercepts deal
52
00:03:37.409 --> 00:03:38.780 A:middle L:90%
with up here. So this is the ex Intercepts
53
00:03:38.780 --> 00:03:42.020 A:middle L:90%
will be set that equal zero so he'd have X
54
00:03:42.020 --> 00:03:46.469 A:middle L:90%
to the fourth plus one is zero. Subtract one
55
00:03:46.520 --> 00:03:49.479 A:middle L:90%
over. We get X to the fourth is equal
56
00:03:49.479 --> 00:03:52.800 A:middle L:90%
to negative one. Take the fourth route on each
57
00:03:52.800 --> 00:03:55.680 A:middle L:90%
side so we get plus or minus+34 through of
58
00:03:55.680 --> 00:04:01.099 A:middle L:90%
negative one. But this year is in imaginary number
59
00:04:01.110 --> 00:04:05.509 A:middle L:90%
since we're taking it even root of a negative number
60
00:04:05.520 --> 00:04:13.819 A:middle L:90%
. So that means we have no ex intercepts and
61
00:04:13.819 --> 00:04:19.100 A:middle L:90%
then to find our horizontal Assam totes, remember or
62
00:04:19.100 --> 00:04:23.560 A:middle L:90%
not horizontal vertical are vertical ascent oats stills with setting
63
00:04:23.569 --> 00:04:25.750 A:middle L:90%
the denominator equal to zero. So we're going to
64
00:04:25.750 --> 00:04:29.250 A:middle L:90%
do X is equal to zero, so that seems
65
00:04:29.250 --> 00:04:31.790 A:middle L:90%
pretty straight or nothing to really solved for right now
66
00:04:31.790 --> 00:04:35.050 A:middle L:90%
, let's use this information or the next part.
67
00:04:36.339 --> 00:04:39.180 A:middle L:90%
So let's just write down right now. All we
68
00:04:39.180 --> 00:04:40.899 A:middle L:90%
know is that we have a vertical awesome toe at
69
00:04:40.980 --> 00:04:43.560 A:middle L:90%
X is equal to zero, and we have no
70
00:04:43.660 --> 00:04:47.310 A:middle L:90%
ex intercepts. So let's go ahead and put down
71
00:04:47.310 --> 00:04:51.009 A:middle L:90%
are dotted line for vertical asked him to hear.
72
00:04:53.740 --> 00:04:58.449 A:middle L:90%
So this is at X is equal to zero.
73
00:05:00.100 --> 00:05:05.970 A:middle L:90%
And now what we are gonna have to do is
74
00:05:06.420 --> 00:05:11.350 A:middle L:90%
determine what side of this we need to start.
75
00:05:13.040 --> 00:05:16.279 A:middle L:90%
So we need to look at. So we had
76
00:05:16.360 --> 00:05:21.560 A:middle L:90%
extra the fourth plus one over X. So let's
77
00:05:21.560 --> 00:05:24.930 A:middle L:90%
look at the limit of this as we approach zero
78
00:05:24.930 --> 00:05:30.209 A:middle L:90%
from the right. So the limits the limit as
79
00:05:30.220 --> 00:05:34.860 A:middle L:90%
X approaches zero from the right. So we're gonna
80
00:05:34.860 --> 00:05:39.029 A:middle L:90%
have zero from the right to the fourth power,
81
00:05:39.029 --> 00:05:42.079 A:middle L:90%
plus one over zero from the right. Well,
82
00:05:42.509 --> 00:05:46.139 A:middle L:90%
if we raise anything that's slightly positive, which means
83
00:05:46.139 --> 00:05:50.329 A:middle L:90%
coming from the right to and even power, that
84
00:05:50.329 --> 00:05:55.259 A:middle L:90%
should still be positive. So we're going to have
85
00:05:56.000 --> 00:05:58.680 A:middle L:90%
zero from the right, plus one over zero from
86
00:05:58.680 --> 00:06:00.649 A:middle L:90%
the right and then adding something slightly to the right
87
00:06:01.139 --> 00:06:06.680 A:middle L:90%
of one of zero and one well that should still
88
00:06:06.680 --> 00:06:09.829 A:middle L:90%
give us something positive and then dividing that by something
89
00:06:09.829 --> 00:06:13.959 A:middle L:90%
positive gives us infinity. So we know that the
90
00:06:13.959 --> 00:06:16.019 A:middle L:90%
right hand side of this will look something kind of
91
00:06:16.019 --> 00:06:20.519 A:middle L:90%
like this. So it was going to start coming
92
00:06:20.519 --> 00:06:24.949 A:middle L:90%
from positivity. And now this is when we use
93
00:06:24.949 --> 00:06:27.579 A:middle L:90%
the fact that we know this has asked Antarctic behavior
94
00:06:27.589 --> 00:06:30.569 A:middle L:90%
like X cubed, so is going to have to
95
00:06:30.579 --> 00:06:33.899 A:middle L:90%
rebound up and then start to fall this green line
96
00:06:35.180 --> 00:06:41.839 A:middle L:90%
up to positive infinity. And then we could do
97
00:06:41.839 --> 00:06:46.560 A:middle L:90%
the same thing to find the limit as exit purchaser
98
00:06:46.560 --> 00:06:49.649 A:middle L:90%
from left. Or we can use the fact that
99
00:06:50.589 --> 00:06:54.180 A:middle L:90%
this is like one over X and we know that
100
00:06:54.180 --> 00:06:58.550 A:middle L:90%
one of Rex has opposite in behavior. So that
101
00:06:58.550 --> 00:07:00.100 A:middle L:90%
means we're gonna have to start from negative infinity over
102
00:07:00.100 --> 00:07:02.569 A:middle L:90%
here, and then we're gonna just keep on going
103
00:07:02.569 --> 00:07:04.480 A:middle L:90%
up. There's no X intercepts or anything. So
104
00:07:04.480 --> 00:07:06.430 A:middle L:90%
then it's just going to turn around and then start
105
00:07:06.439 --> 00:07:11.550 A:middle L:90%
following this green line like that. So this is
106
00:07:11.550 --> 00:07:14.019 A:middle L:90%
a nice way for us to go about getting a
107
00:07:14.019 --> 00:07:15.959 A:middle L:90%
sketch of the graph without doing everything that we've done
108
00:07:15.959 --> 00:07:17.089 A:middle L:90%
in this chapter so far,