Objective: Graph rational functions. Identify slant asymptotes.
In Sections 2.2 and 2.4, we investigated infinite limits and vertical asymptotes. There, we let x...
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Transcript of In Sections 2.2 and 2.4, we investigated infinite limits and vertical asymptotes. There, we let x...
![Page 1: In Sections 2.2 and 2.4, we investigated infinite limits and vertical asymptotes. There, we let x approach a number. The result was that the values.](https://reader035.fdocuments.in/reader035/viewer/2022062308/56649ed95503460f94be88d6/html5/thumbnails/1.jpg)
In Sections 2.2 and 2.4, we
investigated infinite limits and
vertical asymptotes.
There, we let x approach a number.
The result was that the values of y became arbitrarily large (positive or negative).
APPLICATIONS OF DIFFERENTIATION
![Page 2: In Sections 2.2 and 2.4, we investigated infinite limits and vertical asymptotes. There, we let x approach a number. The result was that the values.](https://reader035.fdocuments.in/reader035/viewer/2022062308/56649ed95503460f94be88d6/html5/thumbnails/2.jpg)
In this section, we let become x
arbitrarily large (positive or negative)
and see what happens to y.
We will find it very useful to consider this so-called end behavior when sketching graphs.
APPLICATIONS OF DIFFERENTIATION
![Page 3: In Sections 2.2 and 2.4, we investigated infinite limits and vertical asymptotes. There, we let x approach a number. The result was that the values.](https://reader035.fdocuments.in/reader035/viewer/2022062308/56649ed95503460f94be88d6/html5/thumbnails/3.jpg)
4.4Limits at Infinity;
Horizontal Asymptotes
In this section, we will learn about:
Various aspects of horizontal asymptotes.
APPLICATIONS OF DIFFERENTIATION
![Page 4: In Sections 2.2 and 2.4, we investigated infinite limits and vertical asymptotes. There, we let x approach a number. The result was that the values.](https://reader035.fdocuments.in/reader035/viewer/2022062308/56649ed95503460f94be88d6/html5/thumbnails/4.jpg)
Let’s begin by investigating the behavior
of the function f defined by
as x becomes large.
2
2
1( )
1
xf x
x
HORIZONTAL ASYMPTOTES
![Page 5: In Sections 2.2 and 2.4, we investigated infinite limits and vertical asymptotes. There, we let x approach a number. The result was that the values.](https://reader035.fdocuments.in/reader035/viewer/2022062308/56649ed95503460f94be88d6/html5/thumbnails/5.jpg)
The table gives values of this
function correct to six decimal
places.
The graph of f has been
drawn by a computer in the
figure.
HORIZONTAL ASYMPTOTES
![Page 6: In Sections 2.2 and 2.4, we investigated infinite limits and vertical asymptotes. There, we let x approach a number. The result was that the values.](https://reader035.fdocuments.in/reader035/viewer/2022062308/56649ed95503460f94be88d6/html5/thumbnails/6.jpg)
As x grows larger and larger,
you can see that the values of
f(x) get closer and closer to 1. It seems that we can make the
values of f(x) as close as we like to 1 by taking x sufficiently large.
HORIZONTAL ASYMPTOTES
![Page 7: In Sections 2.2 and 2.4, we investigated infinite limits and vertical asymptotes. There, we let x approach a number. The result was that the values.](https://reader035.fdocuments.in/reader035/viewer/2022062308/56649ed95503460f94be88d6/html5/thumbnails/7.jpg)
This situation is expressed symbolically
by writing
In general, we use the notation
to indicate that the values of f(x) become
closer and closer to L as x becomes larger
and larger.
lim ( )x
f x L
HORIZONTAL ASYMPTOTES
2
2
1lim 1
1x
x
x
![Page 8: In Sections 2.2 and 2.4, we investigated infinite limits and vertical asymptotes. There, we let x approach a number. The result was that the values.](https://reader035.fdocuments.in/reader035/viewer/2022062308/56649ed95503460f94be88d6/html5/thumbnails/8.jpg)
Let f be a function defined on some
interval .
Then,
means that the values of f(x) can be
made arbitrarily close to L by taking x
sufficiently large.
( , )a
lim ( )x
f x L
HORIZONTAL ASYMPTOTES 1. Definition
![Page 9: In Sections 2.2 and 2.4, we investigated infinite limits and vertical asymptotes. There, we let x approach a number. The result was that the values.](https://reader035.fdocuments.in/reader035/viewer/2022062308/56649ed95503460f94be88d6/html5/thumbnails/9.jpg)
Geometric illustrations of Definition 1
are shown in the figures. Notice that there are many ways for the graph of f to
approach the line y = L (which is called a horizontal asymptote) as we look to the far right of each graph.
HORIZONTAL ASYMPTOTES
![Page 10: In Sections 2.2 and 2.4, we investigated infinite limits and vertical asymptotes. There, we let x approach a number. The result was that the values.](https://reader035.fdocuments.in/reader035/viewer/2022062308/56649ed95503460f94be88d6/html5/thumbnails/10.jpg)
Referring to the earlier figure, we see that,
for numerically large negative values of x,
the values of f(x) are close to 1. By letting x decrease through negative values without
bound, we can make f(x) as close as we like to 1.
HORIZONTAL ASYMPTOTES
![Page 11: In Sections 2.2 and 2.4, we investigated infinite limits and vertical asymptotes. There, we let x approach a number. The result was that the values.](https://reader035.fdocuments.in/reader035/viewer/2022062308/56649ed95503460f94be88d6/html5/thumbnails/11.jpg)
This is expressed by writing
The general definition is as follows.
2
2
1lim 1
1x
x
x
HORIZONTAL ASYMPTOTES
![Page 12: In Sections 2.2 and 2.4, we investigated infinite limits and vertical asymptotes. There, we let x approach a number. The result was that the values.](https://reader035.fdocuments.in/reader035/viewer/2022062308/56649ed95503460f94be88d6/html5/thumbnails/12.jpg)
Let f be a function defined on some
interval .
Then,
means that the values of f(x) can be
made arbitrarily close to L by taking x
sufficiently large negative.
( , )a
lim ( )x
f x L
HORIZONTAL ASYMPTOTES 2. Definition
![Page 13: In Sections 2.2 and 2.4, we investigated infinite limits and vertical asymptotes. There, we let x approach a number. The result was that the values.](https://reader035.fdocuments.in/reader035/viewer/2022062308/56649ed95503460f94be88d6/html5/thumbnails/13.jpg)
Again, the symbol does not
represent a number.
However, the expression
is often read as:
“the limit of f(x), as x approaches
negative infinity, is L”
lim ( )x
f x L
HORIZONTAL ASYMPTOTES
![Page 14: In Sections 2.2 and 2.4, we investigated infinite limits and vertical asymptotes. There, we let x approach a number. The result was that the values.](https://reader035.fdocuments.in/reader035/viewer/2022062308/56649ed95503460f94be88d6/html5/thumbnails/14.jpg)
Definition 2
is illustrated in
the figure. Notice that the graph
approaches the line y = L as we look to the far left of each graph.
HORIZONTAL ASYMPTOTES
![Page 15: In Sections 2.2 and 2.4, we investigated infinite limits and vertical asymptotes. There, we let x approach a number. The result was that the values.](https://reader035.fdocuments.in/reader035/viewer/2022062308/56649ed95503460f94be88d6/html5/thumbnails/15.jpg)
The line y = L is called a horizontal
asymptote of the curve y = f(x) if either
lim ( ) or lim ( )x x
f x L f x L
HORIZONTAL ASYMPTOTES 3. Definition
![Page 16: In Sections 2.2 and 2.4, we investigated infinite limits and vertical asymptotes. There, we let x approach a number. The result was that the values.](https://reader035.fdocuments.in/reader035/viewer/2022062308/56649ed95503460f94be88d6/html5/thumbnails/16.jpg)
For instance, the curve illustrated in
the earlier figure has the line y = 1 as
a horizontal asymptote because2
2
1lim 1
1x
x
x
HORIZONTAL ASYMPTOTES 3. Definition
![Page 17: In Sections 2.2 and 2.4, we investigated infinite limits and vertical asymptotes. There, we let x approach a number. The result was that the values.](https://reader035.fdocuments.in/reader035/viewer/2022062308/56649ed95503460f94be88d6/html5/thumbnails/17.jpg)
The curve y = f(x) sketched here has both
y = -1 and y = 2 as horizontal asymptotes.
This is because:
HORIZONTAL ASYMPTOTES
lim 1 and lim 2x x
f x f x
![Page 18: In Sections 2.2 and 2.4, we investigated infinite limits and vertical asymptotes. There, we let x approach a number. The result was that the values.](https://reader035.fdocuments.in/reader035/viewer/2022062308/56649ed95503460f94be88d6/html5/thumbnails/18.jpg)
Find the infinite limits, limits at infinity,
and asymptotes for the function f whose
graph is shown in the figure.
HORIZONTAL ASYMPTOTES Example 1
![Page 19: In Sections 2.2 and 2.4, we investigated infinite limits and vertical asymptotes. There, we let x approach a number. The result was that the values.](https://reader035.fdocuments.in/reader035/viewer/2022062308/56649ed95503460f94be88d6/html5/thumbnails/19.jpg)
We see that the values of f(x) become
large as from both sides.
So,
1x
limx 1
f (x)
HORIZONTAL ASYMPTOTES Example 1
![Page 20: In Sections 2.2 and 2.4, we investigated infinite limits and vertical asymptotes. There, we let x approach a number. The result was that the values.](https://reader035.fdocuments.in/reader035/viewer/2022062308/56649ed95503460f94be88d6/html5/thumbnails/20.jpg)
Notice that f(x) becomes large negative
as x approaches 2 from the left, but large
positive as x approaches 2 from the right. So,
Thus, both the lines x = -1 and x = 2 are vertical asymptotes.
2 2lim ( ) and lim ( )x x
f x f x
HORIZONTAL ASYMPTOTES Example 1
![Page 21: In Sections 2.2 and 2.4, we investigated infinite limits and vertical asymptotes. There, we let x approach a number. The result was that the values.](https://reader035.fdocuments.in/reader035/viewer/2022062308/56649ed95503460f94be88d6/html5/thumbnails/21.jpg)
As x becomes large, it appears that f(x)
approaches 4.
However, as x decreases through negative
values, f(x) approaches 2. So,
and
This means that both y = 4 and y = 2 are horizontal asymptotes.
lim ( ) 4x
f x
HORIZONTAL ASYMPTOTES Example 1
lim ( ) 2x
f x
![Page 22: In Sections 2.2 and 2.4, we investigated infinite limits and vertical asymptotes. There, we let x approach a number. The result was that the values.](https://reader035.fdocuments.in/reader035/viewer/2022062308/56649ed95503460f94be88d6/html5/thumbnails/22.jpg)
Find and
Observe that, when x is large, 1/x is small. For instance,
In fact, by taking x large enough, we can make 1/x as close to 0 as we please.
Therefore, according to Definition 1, we have
1limx x
1limx x
1 1 10.01 , 0.0001 , 0.000001
100 10,000 1,000,000
HORIZONTAL ASYMPTOTES Example 2
1lim 0x x
![Page 23: In Sections 2.2 and 2.4, we investigated infinite limits and vertical asymptotes. There, we let x approach a number. The result was that the values.](https://reader035.fdocuments.in/reader035/viewer/2022062308/56649ed95503460f94be88d6/html5/thumbnails/23.jpg)
Similar reasoning shows that, when x
is large negative, 1/x is small negative.
So, we also have It follows that the line y = 0 (the x-axis) is a horizontal
asymptote of the curve y = 1/x. This is an equilateral hyperbola.
1lim 0x x
HORIZONTAL ASYMPTOTES Example 2
![Page 24: In Sections 2.2 and 2.4, we investigated infinite limits and vertical asymptotes. There, we let x approach a number. The result was that the values.](https://reader035.fdocuments.in/reader035/viewer/2022062308/56649ed95503460f94be88d6/html5/thumbnails/24.jpg)
If r > 0 is a rational number, then
If r > 0 is a rational number such that xr
is defined for all x, then
1lim 0
rx x
1lim 0
rx x
HORIZONTAL ASYMPTOTES 5. Theorem
![Page 25: In Sections 2.2 and 2.4, we investigated infinite limits and vertical asymptotes. There, we let x approach a number. The result was that the values.](https://reader035.fdocuments.in/reader035/viewer/2022062308/56649ed95503460f94be88d6/html5/thumbnails/25.jpg)
Evaluate
and indicate which properties of limits
are used at each stage.
As x becomes large, both numerator and denominator become large.
So, it isn’t obvious what happens to their ratio. We need to do some preliminary algebra.
2
2
3 2lim
5 4 1x
x x
x x
HORIZONTAL ASYMPTOTES Example 3
![Page 26: In Sections 2.2 and 2.4, we investigated infinite limits and vertical asymptotes. There, we let x approach a number. The result was that the values.](https://reader035.fdocuments.in/reader035/viewer/2022062308/56649ed95503460f94be88d6/html5/thumbnails/26.jpg)
To evaluate the limit at infinity of any rational
function, we first divide both the numerator
and denominator by the highest power of x
that occurs in the denominator. We may assume that , since we are interested
in only large values of x.0x
HORIZONTAL ASYMPTOTES Example 3
![Page 27: In Sections 2.2 and 2.4, we investigated infinite limits and vertical asymptotes. There, we let x approach a number. The result was that the values.](https://reader035.fdocuments.in/reader035/viewer/2022062308/56649ed95503460f94be88d6/html5/thumbnails/27.jpg)
In this case, the highest power of x in the
denominator is x2. So, we have:2
2 2 2
2 2
22
3 2 1 23
3 2lim lim lim
4 15 4 1 5 4 1 5x x x
x xx x xx xx x x x
x xx
HORIZONTAL ASYMPTOTES Example 3
![Page 28: In Sections 2.2 and 2.4, we investigated infinite limits and vertical asymptotes. There, we let x approach a number. The result was that the values.](https://reader035.fdocuments.in/reader035/viewer/2022062308/56649ed95503460f94be88d6/html5/thumbnails/28.jpg)
3 0 0(by Limit Law 7 and Theoreom 5)
5 0 03
5
2
2
1 1lim3 lim 2lim
(by Limit Laws 1, 2, and 3)1 1
lim5 4 lim lim
x x x
x x x
x x
x x
HORIZONTAL ASYMPTOTES Example 3
2
2
1 2lim 3
(by Limit Law 5)4 1
lim 5
x
x
x x
x x
![Page 29: In Sections 2.2 and 2.4, we investigated infinite limits and vertical asymptotes. There, we let x approach a number. The result was that the values.](https://reader035.fdocuments.in/reader035/viewer/2022062308/56649ed95503460f94be88d6/html5/thumbnails/29.jpg)
A similar calculation shows that the limit
as is also
The figure illustrates the results of these calculations by showing how the graph of the given rational function approaches the horizontal asymptote
x 3
5
3
5y
HORIZONTAL ASYMPTOTES Example 3
![Page 30: In Sections 2.2 and 2.4, we investigated infinite limits and vertical asymptotes. There, we let x approach a number. The result was that the values.](https://reader035.fdocuments.in/reader035/viewer/2022062308/56649ed95503460f94be88d6/html5/thumbnails/30.jpg)
Find the horizontal and vertical
asymptotes of the graph of the
function22 1
( )3 5
xf x
x
HORIZONTAL ASYMPTOTES Example 4
![Page 31: In Sections 2.2 and 2.4, we investigated infinite limits and vertical asymptotes. There, we let x approach a number. The result was that the values.](https://reader035.fdocuments.in/reader035/viewer/2022062308/56649ed95503460f94be88d6/html5/thumbnails/31.jpg)
Dividing both numerator and denominator
by x and using the properties of limits,
we have:
HORIZONTAL ASYMPTOTES Example 4
2 22
12
2 1lim lim (since for 0)
53 5 3x x
x x x x xx
x
2 2
1 1lim 2 lim 2 lim
2 0 215 3 5.0 3lim3 5limlim 3
x x x
x xx
x x
xx
![Page 32: In Sections 2.2 and 2.4, we investigated infinite limits and vertical asymptotes. There, we let x approach a number. The result was that the values.](https://reader035.fdocuments.in/reader035/viewer/2022062308/56649ed95503460f94be88d6/html5/thumbnails/32.jpg)
Therefore, the line is
a horizontal asymptote of the graph of f.
2 / 3y HORIZONTAL ASYMPTOTES Example 4
![Page 33: In Sections 2.2 and 2.4, we investigated infinite limits and vertical asymptotes. There, we let x approach a number. The result was that the values.](https://reader035.fdocuments.in/reader035/viewer/2022062308/56649ed95503460f94be88d6/html5/thumbnails/33.jpg)
In computing the limit as ,
we must remember that, for x < 0,
we have
So, when we divide the numerator by x, for x < 0, we get
Therefore,
x
2x x x
2 222
1 1 12 1 2 1 2x x
x xx
limx
2x2 1
3x 5 lim
x
2 1
x2
3 5
x
2 lim
x
1
x2
3 5 limx
1
x
2
3
HORIZONTAL ASYMPTOTES Example 4
![Page 34: In Sections 2.2 and 2.4, we investigated infinite limits and vertical asymptotes. There, we let x approach a number. The result was that the values.](https://reader035.fdocuments.in/reader035/viewer/2022062308/56649ed95503460f94be88d6/html5/thumbnails/34.jpg)
Thus, the line is also
a horizontal asymptote.
23y
HORIZONTAL ASYMPTOTES Example 4
![Page 35: In Sections 2.2 and 2.4, we investigated infinite limits and vertical asymptotes. There, we let x approach a number. The result was that the values.](https://reader035.fdocuments.in/reader035/viewer/2022062308/56649ed95503460f94be88d6/html5/thumbnails/35.jpg)
A vertical asymptote is likely to occur
when the denominator, 3x - 5, is 0,
that is, when
If x is close to and , then the denominator is close to 0 and 3x - 5 is positive.
The numerator is always positive, so f(x) is positive.
Therefore,
5
3x 5
35
3x
22 1x
HORIZONTAL ASYMPTOTES Example 4
2
(5 3)
2 1lim
3 5x
x
x
![Page 36: In Sections 2.2 and 2.4, we investigated infinite limits and vertical asymptotes. There, we let x approach a number. The result was that the values.](https://reader035.fdocuments.in/reader035/viewer/2022062308/56649ed95503460f94be88d6/html5/thumbnails/36.jpg)
If x is close to but , then 3x – 5 < 0, so f(x) is large negative.
Thus,
The vertical asymptote is
5
35
3x
2
(5 3)
2 1lim
3 5x
x
x
5
3x
HORIZONTAL ASYMPTOTES Example 4
![Page 37: In Sections 2.2 and 2.4, we investigated infinite limits and vertical asymptotes. There, we let x approach a number. The result was that the values.](https://reader035.fdocuments.in/reader035/viewer/2022062308/56649ed95503460f94be88d6/html5/thumbnails/37.jpg)
Compute
As both and x are large when x is large, it’s difficult to see what happens to their difference.
So, we use algebra to rewrite the function.
2lim 1x
x x
2 1x
HORIZONTAL ASYMPTOTES Example 5
![Page 38: In Sections 2.2 and 2.4, we investigated infinite limits and vertical asymptotes. There, we let x approach a number. The result was that the values.](https://reader035.fdocuments.in/reader035/viewer/2022062308/56649ed95503460f94be88d6/html5/thumbnails/38.jpg)
We first multiply the numerator and
denominator by the conjugate radical:
2
2 2
2
2 2
2 2
1lim 1 lim 1
1
( 1) 1lim lim
1 1
x x
x x
x xx x x x
x x
x x
x x x x
HORIZONTAL ASYMPTOTES Example 5
![Page 39: In Sections 2.2 and 2.4, we investigated infinite limits and vertical asymptotes. There, we let x approach a number. The result was that the values.](https://reader035.fdocuments.in/reader035/viewer/2022062308/56649ed95503460f94be88d6/html5/thumbnails/39.jpg)
However, an easier method is to divide the numerator and denominator by x.
Doing this and using the Limit Laws, we obtain:
2
2 2
2
11
lim 1 lim lim1 1
10
lim 01 1 0 1
1 1
x x x
x
xx xx x x x
x
x
x
HORIZONTAL ASYMPTOTES Example 5
![Page 40: In Sections 2.2 and 2.4, we investigated infinite limits and vertical asymptotes. There, we let x approach a number. The result was that the values.](https://reader035.fdocuments.in/reader035/viewer/2022062308/56649ed95503460f94be88d6/html5/thumbnails/40.jpg)
The figure illustrates this
result.
HORIZONTAL ASYMPTOTES Example 5
![Page 41: In Sections 2.2 and 2.4, we investigated infinite limits and vertical asymptotes. There, we let x approach a number. The result was that the values.](https://reader035.fdocuments.in/reader035/viewer/2022062308/56649ed95503460f94be88d6/html5/thumbnails/41.jpg)
Evaluate
If we let t = 1/x, then as .
Therefore, .
1lim sin x x
0t x
0
1lim sin lim sin 0
x tt
x
HORIZONTAL ASYMPTOTES Example 6
![Page 42: In Sections 2.2 and 2.4, we investigated infinite limits and vertical asymptotes. There, we let x approach a number. The result was that the values.](https://reader035.fdocuments.in/reader035/viewer/2022062308/56649ed95503460f94be88d6/html5/thumbnails/42.jpg)
Evaluate
As x increases, the values of sin x oscillate between 1 and -1 infinitely often.
So, they don’t approach any definite number. Thus, does not exist.
limsinx
x
HORIZONTAL ASYMPTOTES Example 7
limsinx
x
![Page 43: In Sections 2.2 and 2.4, we investigated infinite limits and vertical asymptotes. There, we let x approach a number. The result was that the values.](https://reader035.fdocuments.in/reader035/viewer/2022062308/56649ed95503460f94be88d6/html5/thumbnails/43.jpg)
The notation is used to
indicate that the values of f(x) become
large as x becomes large. Similar meanings are attached to the following symbols:
lim ( )x
f x
lim ( )x
f x
INFINITE LIMITS AT INFINITY
lim ( )x
f x
lim ( )x
f x
![Page 44: In Sections 2.2 and 2.4, we investigated infinite limits and vertical asymptotes. There, we let x approach a number. The result was that the values.](https://reader035.fdocuments.in/reader035/viewer/2022062308/56649ed95503460f94be88d6/html5/thumbnails/44.jpg)
Find and
When x becomes large, x3 also becomes large.
For instance,
In fact, we can make x3 as big as we like by taking x large enough.
Therefore, we can write
3limx
x
3limx
x
3 3 310 1,000 100 1,000,000 1,000 1,000,000,000
3limx
x
Example 8INFINITE LIMITS AT INFINITY
![Page 45: In Sections 2.2 and 2.4, we investigated infinite limits and vertical asymptotes. There, we let x approach a number. The result was that the values.](https://reader035.fdocuments.in/reader035/viewer/2022062308/56649ed95503460f94be88d6/html5/thumbnails/45.jpg)
Similarly, when x is large negative, so is x3. Thus,
These limit statements can also be seen from the graph of y = x3 in the figure.
3limx
x
Example 8INFINITE LIMITS AT INFINITY
![Page 46: In Sections 2.2 and 2.4, we investigated infinite limits and vertical asymptotes. There, we let x approach a number. The result was that the values.](https://reader035.fdocuments.in/reader035/viewer/2022062308/56649ed95503460f94be88d6/html5/thumbnails/46.jpg)
Find
It would be wrong to write
The Limit Laws can’t be applied to infinite limits because is not a number ( can’t be defined).
However, we can write
This is because both x and x - 1 become arbitrarily large and so their product does too.
2lim( )x
x x
2 2lim( ) lim limx x x
x x x x
2lim( ) lim ( 1)x x
x x x x
Example 9INFINITE LIMITS AT INFINITY
![Page 47: In Sections 2.2 and 2.4, we investigated infinite limits and vertical asymptotes. There, we let x approach a number. The result was that the values.](https://reader035.fdocuments.in/reader035/viewer/2022062308/56649ed95503460f94be88d6/html5/thumbnails/47.jpg)
Find
As in Example 3, we divide the numerator and denominator by the highest power of x in the denominator, which is just x:
because and as
2
lim3x
x x
x
2 1lim lim
33 1x x
x x x
xx
1x 3 1 1x x
Example 10INFINITE LIMITS AT INFINITY
![Page 48: In Sections 2.2 and 2.4, we investigated infinite limits and vertical asymptotes. There, we let x approach a number. The result was that the values.](https://reader035.fdocuments.in/reader035/viewer/2022062308/56649ed95503460f94be88d6/html5/thumbnails/48.jpg)
The next example shows that, by using
infinite limits at infinity, together with
intercepts, we can get a rough idea of the
graph of a polynomial without computingderivatives.
INFINITE LIMITS AT INFINITY
![Page 49: In Sections 2.2 and 2.4, we investigated infinite limits and vertical asymptotes. There, we let x approach a number. The result was that the values.](https://reader035.fdocuments.in/reader035/viewer/2022062308/56649ed95503460f94be88d6/html5/thumbnails/49.jpg)
Sketch the graph of
by finding its intercepts and its limits
as and as
The y-intercept is f(0) = (-2)4(1)3(-1) = -16 The x-intercepts are found by setting y = 0: x = 2, -1, 1.
4 3( 2) ( 1) ( 1)y x x x
x x
Example 11INFINITE LIMITS AT INFINITY
![Page 50: In Sections 2.2 and 2.4, we investigated infinite limits and vertical asymptotes. There, we let x approach a number. The result was that the values.](https://reader035.fdocuments.in/reader035/viewer/2022062308/56649ed95503460f94be88d6/html5/thumbnails/50.jpg)
Notice that, since (x - 2)4 is positive,
the function doesn’t change sign at 2.
Thus, the graph doesn’t cross the x-axis
at 2. It crosses the axis at -1 and 1.
Example 11INFINITE LIMITS AT INFINITY
![Page 51: In Sections 2.2 and 2.4, we investigated infinite limits and vertical asymptotes. There, we let x approach a number. The result was that the values.](https://reader035.fdocuments.in/reader035/viewer/2022062308/56649ed95503460f94be88d6/html5/thumbnails/51.jpg)
When x is large positive, all three factors
are large, so
When x is large negative, the first factor
is large positive and the second and third
factors are both large negative, so4 3lim ( 2) ( 1) ( 1)
xx x x
4 3lim( 2) ( 1) ( 1)x
x x x
Example 11INFINITE LIMITS AT INFINITY
![Page 52: In Sections 2.2 and 2.4, we investigated infinite limits and vertical asymptotes. There, we let x approach a number. The result was that the values.](https://reader035.fdocuments.in/reader035/viewer/2022062308/56649ed95503460f94be88d6/html5/thumbnails/52.jpg)
Combining this information,
we give a rough sketch of the graph
in the figure.
Example 11INFINITE LIMITS AT INFINITY
![Page 53: In Sections 2.2 and 2.4, we investigated infinite limits and vertical asymptotes. There, we let x approach a number. The result was that the values.](https://reader035.fdocuments.in/reader035/viewer/2022062308/56649ed95503460f94be88d6/html5/thumbnails/53.jpg)
In Example 3, we calculated that
In the next example, we use
a graphing device to relate this statement
to Definition 5 with and .
2
2
3 2 3lim
5 4 1 5x
x x
x x
3
5L 0.1
PRECISE DEFINITIONS
![Page 54: In Sections 2.2 and 2.4, we investigated infinite limits and vertical asymptotes. There, we let x approach a number. The result was that the values.](https://reader035.fdocuments.in/reader035/viewer/2022062308/56649ed95503460f94be88d6/html5/thumbnails/54.jpg)
Use a graph to find a number N
such that, if x > N, then
We rewrite the given inequality as:
PRECISE DEFINITIONS Example 12
2
2
3 20.6 0.1
5 4 1
x x
x x
2
2
3 20.5 0.7
5 4 1
x x
x x
![Page 55: In Sections 2.2 and 2.4, we investigated infinite limits and vertical asymptotes. There, we let x approach a number. The result was that the values.](https://reader035.fdocuments.in/reader035/viewer/2022062308/56649ed95503460f94be88d6/html5/thumbnails/55.jpg)
We need to determine the values of x
for which the given curve lies between
the horizontal lines y = 0.5 and y = 0.7 So, we graph the curve and
these lines in the figure.
PRECISE DEFINITIONS Example 12
![Page 56: In Sections 2.2 and 2.4, we investigated infinite limits and vertical asymptotes. There, we let x approach a number. The result was that the values.](https://reader035.fdocuments.in/reader035/viewer/2022062308/56649ed95503460f94be88d6/html5/thumbnails/56.jpg)
Then, we use the cursor to estimate
that the curve crosses the line y = 0.5
when To the right of this number, the curve stays between
the lines y = 0.5 and y = 0.7
6.7x
PRECISE DEFINITIONS Example 12
![Page 57: In Sections 2.2 and 2.4, we investigated infinite limits and vertical asymptotes. There, we let x approach a number. The result was that the values.](https://reader035.fdocuments.in/reader035/viewer/2022062308/56649ed95503460f94be88d6/html5/thumbnails/57.jpg)
Rounding to be safe, we can say that,
if x > 7, then
In other words, for , we can choose N = 7 (or any larger number) in Definition 5.
2
2
3 20.6 0.1
5 4 1
x x
x x
0.1
PRECISE DEFINITIONS Example 12