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.
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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
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
4.4Limits at Infinity;
Horizontal Asymptotes
In this section, we will learn about:
Various aspects of horizontal asymptotes.
APPLICATIONS OF DIFFERENTIATION
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
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
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
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
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
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
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
This is expressed by writing
The general definition is as follows.
2
2
1lim 1
1x
x
x
HORIZONTAL ASYMPTOTES
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
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
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
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
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
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
Find the infinite limits, limits at infinity,
and asymptotes for the function f whose
graph is shown in the figure.
HORIZONTAL ASYMPTOTES Example 1
We see that the values of f(x) become
large as from both sides.
So,
1x
limx 1
f (x)
HORIZONTAL ASYMPTOTES Example 1
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
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
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
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
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
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
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
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
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
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
Find the horizontal and vertical
asymptotes of the graph of the
function22 1
( )3 5
xf x
x
HORIZONTAL ASYMPTOTES Example 4
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
Therefore, the line is
a horizontal asymptote of the graph of f.
2 / 3y HORIZONTAL ASYMPTOTES Example 4
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
Thus, the line is also
a horizontal asymptote.
23y
HORIZONTAL ASYMPTOTES Example 4
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
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
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
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
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
The figure illustrates this
result.
HORIZONTAL ASYMPTOTES Example 5
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
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
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
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
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
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
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
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
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
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
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
Combining this information,
we give a rough sketch of the graph
in the figure.
Example 11INFINITE LIMITS AT INFINITY
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
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
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
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
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
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