### Limits at Infinity; Horizontal Asymptotes

*In this section we let x become arbitrarily large (positive or *
*negative) and see what happens to y.*

*Let’s begin by investigating the behavior of the function f*
defined by

*as x becomes large.*

2

### Limits at Infinity; Horizontal Asymptotes

The table gives values of this function correct to six decimal
*places, and the graph of f has been drawn by a computer in *
Figure 1.

**Figure 1**

### Limits at Infinity; Horizontal Asymptotes

*As x grows larger and larger you can see that the values of*
*f(x) get closer and closer to 1. In fact, it seems that we can *
*make the values of f(x) as close as we like to 1 by taking x*
sufficiently large.

This situation is expressed symbolically by writing

4

### Limits at Infinity; Horizontal Asymptotes

In general, we use the notation

*to indicate that the values of f(x) approach L as x becomes *
larger and larger.

### Limits at Infinity; Horizontal Asymptotes

Another notation for is
*f(x) → L as x →*

Geometric illustrations of Definition 1 are shown in Figure 2.

6

### Limits at Infinity; Horizontal Asymptotes

*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.*

### Limits at Infinity; Horizontal Asymptotes

Referring back to Figure 1, 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 to 1 as we like.*

This is expressed by writing

**Figure 1**

8

### Limits at Infinity; Horizontal Asymptotes

The general definition is as follows.

### Limits at Infinity; Horizontal Asymptotes

Definition 2 is illustrated in Figure 3. Notice that the graph
*approaches the line y = L as we look to the far left of each *
graph.

Examples illustrating

**Figure 3**

10

### Limits at Infinity; Horizontal Asymptotes

## Example 2

Find and Solution:

*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.

12

*Example 2 – Solution*

Therefore, according to Definition 1, we have

= 0

*Similar reasoning shows that when x is large negative, 1/x*
is small negative, so we also have

= 0

cont’d

*Example 2 – Solution*

*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; see Figure 6.)

**Figure 6**

cont’d

14

### Limits at Infinity; Horizontal Asymptotes

## Example 3

Evaluate and indicate which properties of limits are used at each stage.

Solution:

*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.

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 x ≠ 0, since we are interested only in *

16

*Example 3 – Solution*

*In this case the highest power of x in the denominator is x*^{2},
so we have

cont’d

*Example 3 – Solution*

_{cont’d}

(by Limit Law 5)

(by 1, 2, and 3)

(by 7 and Theorem 4)

18

*Example 3 – Solution*

*A similar calculation shows that the limit as x* → –
is also

Figure 7 illustrates the results of these calculations by showing how the graph of the given

rational function approaches the horizontal asymptote

*y = = 0.6.*

cont’d

**Figure 7**

## Example 4

Find the horizontal and vertical asymptotes of the graph of the function

Solution:

*Dividing both numerator and denominator by x and using *
the properties of limits, we have

20

*Example 4 – Solution*

_{cont’d}

*(since = x for x > 0) *

*Example 4 – Solution*

*Therefore the line y = is a horizontal asymptote of the *
*graph of f.*

*In computing the limit as x → – , we must remember that *
*for x < 0, we have = | x| = –x. *

cont’d

22

*Example 4 – Solution*

*So when we divide the numerator by x, for x < 0 we get*

cont’d

*Example 4 – Solution*

Therefore

cont’d

24

*Example 4 – Solution*

*Thus the line y = –* is also a horizontal asymptote.

A vertical asymptote is likely to occur when the
*denominator, 3x – 5, is 0, that is, when*

*If x is close to and x > , then the denominator is close to *
*0 and 3x – 5 is positive. The numerator is always *
*positive, so f(x) is positive.*

Therefore

cont’d

*Example 4 – Solution*

*If x is close to but x < , then 3x – 5 < 0 and so f(x) is *
large negative. Thus

*The vertical asymptote is x = .*
All three asymptotes are shown
in Figure 8.

cont’d

26

## Infinite Limits at Infinity

## Infinite Limits at Infinity

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:

28

## Example 8

Find and Solution:

*When x becomes large, x*^{3} also becomes large. For
instance,

*In fact, we can make x*^{3} *as big as we like by requiring x to *
be large enough. Therefore we can write

*Example 8 – Solution*

*Similarly, when x is large negative, so is x*^{3}. Thus

These limit statements can also be seen from the graph
*of y = x*^{3 }in Figure 10.

cont’d

30

## Precise Definitions

## Precise Definitions

Definition 1 can be stated precisely as follows.

*In words, this says that the values of f(x) can be made *
*arbitrarily close to L (within a distance ε, where ε is any *
*positive number) by requiring x to be sufficiently large *
*(larger than N, where N depends on ε).*

32

## Precise Definitions

*Graphically it says that by keeping x large enough (larger *
*than some number N) we can make the graph of f lie *

*between the given horizontal lines y = L –* *ε and y = L + ε*
as in Figure 12.

**Figure 12**

## Precise Definitions

This must be true no matter how small we choose *ε. *

Figure 13 shows that if a smaller value of *ε is chosen, then *
*a larger value of N may be required.*

**Figure 13**

34

## Precise Definitions

## Example 13

Use Definition 5 to prove that = 0.

Solution:

Given *ε > 0, we want to find N such that*
*if x > N then*

*In computing the limit we may assume that x > 0.*

Then

*1/x <* *ε* *x > 1/ε.*

36

*Example 13 – Solution*

Let’s choose N = 1/*ε. So*
if then
Therefore, by Definition 5,

= 0

cont’d

*Example 13 – Solution*

Figure 16 illustrates the proof by showing some values of *ε*
*and the corresponding values of N.*

cont’d

**Figure 16**

38

## Precise Definitions

Finally we note that an infinite limit at infinity can be defined as follows. The geometric illustration is given in Figure 17.

**Figure 17**

Similar definitions apply when the symbol is replaced by – .