4. Applications of Derivatives

# 4.6 Limits at Infinity and Asymptotes

### Learning Objectives

- Calculate the limit of a function as increases or decreases without bound.
- Recognize a horizontal asymptote on the graph of a function.
- Estimate the end behavior of a function as increases or decreases without bound.
- Recognize an oblique asymptote on the graph of a function.
- Analyze a function and its derivatives to draw its graph.

We have shown how to use the first and second derivatives of a function to describe the shape of a graph. To graph a function defined on an unbounded domain, we also need to know the behavior of as In this section, we define limits at infinity and show how these limits affect the graph of a function. At the end of this section, we outline a strategy for graphing an arbitrary function

# Limits at Infinity

We begin by examining what it means for a function to have a finite **limit at infinity.** Then we study the idea of a function with an **infinite limit at infinity**. Back in Introduction to Functions and Graphs, we looked at vertical asymptotes; in this section we deal with horizontal and oblique asymptotes.

## Limits at Infinity and Horizontal Asymptotes

Recall that means becomes arbitrarily close to as long as is sufficiently close to We can extend this idea to limits at infinity. For example, consider the function As can be seen graphically in (Figure) and numerically in (Figure), as the values of get larger, the values of approach 2. We say the limit as approaches of is 2 and write Similarly, for as the values get larger, the values of approaches 2. We say the limit as approaches of is 2 and write

10 | 100 | 1,000 | 10,000 | |

2.1 | 2.01 | 2.001 | 2.0001 | |

-10 | -100 | -1000 | -10,000 | |

1.9 | 1.99 | 1.999 | 1.9999 |

More generally, for any function we say the limit as of is if becomes arbitrarily close to as long as is sufficiently large. In that case, we write Similarly, we say the limit as of is if becomes arbitrarily close to as long as and is sufficiently large. In that case, we write We now look at the definition of a function having a limit at infinity.

### Definition

(Informal) If the values of become arbitrarily close to as becomes sufficiently large, we say the function has a limit at infinity and write

If the values of becomes arbitrarily close to for as becomes sufficiently large, we say that the function has a limit at negative infinity and write

If the values are getting arbitrarily close to some finite value as or the graph of approaches the line In that case, the line is a horizontal asymptote of ((Figure)). For example, for the function since the line is a horizontal asymptote of

### Definition

If or we say the line is a **horizontal asymptote** of

A function cannot cross a vertical asymptote because the graph must approach infinity (or from at least one direction as approaches the vertical asymptote. However, a function may cross a horizontal asymptote. In fact, a function may cross a horizontal asymptote an unlimited number of times. For example, the function shown in (Figure) intersects the horizontal asymptote an infinite number of times as it oscillates around the asymptote with ever-decreasing amplitude.

The algebraic limit laws and squeeze theorem we introduced in Introduction to Limits also apply to limits at infinity. We illustrate how to use these laws to compute several limits at infinity.

### Computing Limits at Infinity

For each of the following functions evaluate and Determine the horizontal asymptote(s) for

#### Solution

- Using the algebraic limit laws, we have Similarly, Therefore, has a horizontal asymptote of and approaches this horizontal asymptote as as shown in the following graph.
- Since for all we have
for all Also, since

we can apply the squeeze theorem to conclude that

Similarly,

Thus, has a horizontal asymptote of and approaches this horizontal asymptote as as shown in the following graph.

- To evaluate and we first consider the graph of over the interval as shown in the following graph.

Since

it follows that

Similarly, since

it follows that

As a result, and are horizontal asymptotes of as shown in the following graph.

Evaluate and Determine the horizontal asymptotes of if any.

#### Solution

Both limits are 3. The line is a horizontal asymptote.

## Infinite Limits at Infinity

Sometimes the values of a function become arbitrarily large as (or as In this case, we write (or On the other hand, if the values of are negative but become arbitrarily large in magnitude as (or as we write (or

For example, consider the function As seen in (Figure) and (Figure), as the values become arbitrarily large. Therefore, On the other hand, as the values of are negative but become arbitrarily large in magnitude. Consequently,

10 | 20 | 50 | 100 | 1000 | |

1000 | 8000 | 125,000 | 1,000,000 | 1,000,000,000 | |

-10 | -20 | -50 | -100 | -1000 | |

-1000 | -8000 | -125,000 | -1,000,000 | -1,000,000,000 |

### Definition

(Informal) We say a function has an infinite limit at infinity and write

if becomes arbitrarily large for sufficiently large. We say a function has a negative infinite limit at infinity and write

if and becomes arbitrarily large for sufficiently large. Similarly, we can define infinite limits as

## Formal Definitions

Earlier, we used the terms *arbitrarily close*, *arbitrarily large*, and *sufficiently large* to define limits at infinity informally. Although these terms provide accurate descriptions of limits at infinity, they are not precise mathematically. Here are more formal definitions of limits at infinity. We then look at how to use these definitions to prove results involving limits at infinity.

### Definition

(Formal) We say a function has a limit at infinity, if there exists a real number such that for all there exists such that

for all In that case, we write

(see (Figure)).

We say a function has a limit at negative infinity if there exists a real number such that for all there exists such that

for all In that case, we write

Earlier in this section, we used graphical evidence in (Figure) and numerical evidence in (Figure) to conclude that Here we use the formal definition of limit at infinity to prove this result rigorously.

### A Finite Limit at Infinity Example

Use the formal definition of limit at infinity to prove that

#### Solution

Let Let Therefore, for all we have

Use the formal definition of limit at infinity to prove that

#### Solution

Let Let Therefore, for all we have

Therefore,

#### Hint

Let

We now turn our attention to a more precise definition for an infinite limit at infinity.

### Definition

(Formal) We say a function has an infinite limit at infinity and write

if for all there exists an such that

for all (see (Figure)).

We say a function has a negative infinite limit at infinity and write

if for all there exists an such that

for all

Similarly we can define limits as

Earlier, we used graphical evidence ((Figure)) and numerical evidence ((Figure)) to conclude that Here we use the formal definition of infinite limit at infinity to prove that result.

### An Infinite Limit at Infinity

Use the formal definition of infinite limit at infinity to prove that

#### Solution

Let Let Then, for all we have

Therefore,

Use the formal definition of infinite limit at infinity to prove that

#### Solution

Let Let Then, for all we have

#### Hint

Let

# End Behavior

The behavior of a function as is called the function’s **end behavior**. At each of the function’s ends, the function could exhibit one of the following types of behavior:

- The function approaches a horizontal asymptote
- The function or
- The function does not approach a finite limit, nor does it approach or In this case, the function may have some oscillatory behavior.

Let’s consider several classes of functions here and look at the different types of end behaviors for these functions.

## End Behavior for Polynomial Functions

Consider the power function where is a positive integer. From (Figure) and (Figure), we see that

and

Using these facts, it is not difficult to evaluate and where is any constant and is a positive integer. If the graph of is a vertical stretch or compression of and therefore

If the graph of is a vertical stretch or compression combined with a reflection about the -axis, and therefore

If in which case

### Limits at Infinity for Power Functions

For each function evaluate and

#### Solution

- Since the coefficient of is -5, the graph of involves a vertical stretch and reflection of the graph of about the -axis. Therefore, and
- Since the coefficient of is 2, the graph of is a vertical stretch of the graph of Therefore, and

Let Find

#### Solution

#### Hint

The coefficient -3 is negative.

We now look at how the limits at infinity for power functions can be used to determine for any polynomial function Consider a polynomial function

of degree so that Factoring, we see that

As all the terms inside the parentheses approach zero except the first term. We conclude that

For example, the function behaves like as as shown in (Figure) and (Figure).

10 | 100 | 1000 | |

4704 | 4,970,004 | 4,997,000,004 | |

5000 | 5,000,000 | 5,000,000,000 | |

-10 | -100 | -1000 | |

-5296 | -5,029,996 | -5,002,999,996 | |

-5000 | -5,000,000 | -5,000,000,000 |

## End Behavior for Algebraic Functions

The end behavior for rational functions and functions involving radicals is a little more complicated than for polynomials. In (Figure), we show that the limits at infinity of a rational function depend on the relationship between the degree of the numerator and the degree of the denominator. To evaluate the limits at infinity for a rational function, we divide the numerator and denominator by the highest power of appearing in the denominator. This determines which term in the overall expression dominates the behavior of the function at large values of

### Determining End Behavior for Rational Functions

For each of the following functions, determine the limits as and Then, use this information to describe the end behavior of the function.

- (
*Note:*The degree of the numerator and the denominator are the same.) - (
*Note:*The degree of numerator is less than the degree of the denominator.) - (
*Note:*The degree of numerator is greater than the degree of the denominator.)

#### Solution

- The highest power of in the denominator is Therefore, dividing the numerator and denominator by and applying the algebraic limit laws, we see that
Since we know that is a horizontal asymptote for this function as shown in the following graph.

- Since the largest power of appearing in the denominator is divide the numerator and denominator by After doing so and applying algebraic limit laws, we obtain
Therefore has a horizontal asymptote of as shown in the following graph.

- Dividing the numerator and denominator by we have
As the denominator approaches 1. As the numerator approaches As the numerator approaches Therefore whereas as shown in the following figure.

Evaluate and use these limits to determine the end behavior of

#### Solution

#### Hint

Divide the numerator and denominator by

Before proceeding, consider the graph of shown in (Figure). As and the graph of appears almost linear. Although is certainly not a linear function, we now investigate why the graph of seems to be approaching a linear function. First, using long division of polynomials, we can write

Since as we conclude that

Therefore, the graph of approaches the line as This line is known as an **oblique asymptote** for ((Figure)).

We can summarize the results of (Figure) to make the following conclusion regarding end behavior for rational functions. Consider a rational function

where

- If the degree of the numerator is the same as the degree of the denominator then has a horizontal asymptote of as
- If the degree of the numerator is less than the degree of the denominator then has a horizontal asymptote of as
- If the degree of the numerator is greater than the degree of the denominator then does not have a horizontal asymptote. The limits at infinity are either positive or negative infinity, depending on the signs of the leading terms. In addition, using long division, the function can be rewritten as
where the degree of is less than the degree of As a result, Therefore, the values of approach zero as If the degree of is exactly one more than the degree of the function is a linear function. In this case, we call an oblique asymptote.

Now let’s consider the end behavior for functions involving a radical.

### Determining End Behavior for a Function Involving a Radical

Find the limits as and for and describe the end behavior of

#### Solution

Let’s use the same strategy as we did for rational functions: divide the numerator and denominator by a power of To determine the appropriate power of consider the expression in the denominator. Since

for large values of in effect appears just to the first power in the denominator. Therefore, we divide the numerator and denominator by Then, using the fact that for for and for all we calculate the limits as follows:

Therefore, approaches the horizontal asymptote as and the horizontal asymptote as as shown in the following graph.

Evaluate

#### Solution

#### Hint

Divide the numerator and denominator by

## Determining End Behavior for Transcendental Functions

The six basic trigonometric functions are periodic and do not approach a finite limit as For example, oscillates between ((Figure)). The tangent function has an infinite number of vertical asymptotes as therefore, it does not approach a finite limit nor does it approach as as shown in (Figure).

Recall that for any base the function is an exponential function with domain and range If is increasing over If is decreasing over For the natural exponential function Therefore, is increasing on and the range is The exponential function approaches as and approaches 0 as as shown in (Figure) and (Figure).

-5 | -2 | 0 | 2 | 5 | |

0.00674 | 0.135 | 1 | 7.389 | 148.413 |

Recall that the natural logarithm function is the inverse of the natural exponential function Therefore, the domain of is and the range is The graph of is the reflection of the graph of about the line Therefore, as and as as shown in (Figure) and (Figure).

0.01 | 0.1 | 1 | 10 | 100 | |

-4.605 | -2.303 | 0 | 2.303 | 4.605 |

### Determining End Behavior for a Transcendental Function

Find the limits as and for and describe the end behavior of

#### Solution

To find the limit as divide the numerator and denominator by

As shown in (Figure), as Therefore,

We conclude that and the graph of approaches the horizontal asymptote as To find the limit as use the fact that as to conclude that and therefore the graph of approaches the horizontal asymptote as

Find the limits as and for

#### Solution

[latex]\underset{x\to \text{−}\infty }{\text{lim}}f(x)=-2[/latex]

#### Hint

and

# Guidelines for Drawing the Graph of a Function

We now have enough analytical tools to draw graphs of a wide variety of algebraic and transcendental functions. Before showing how to graph specific functions, let’s look at a general strategy to use when graphing any function.

### Problem-Solving Strategy: Drawing the Graph of a Function

Given a function use the following steps to sketch a graph of

- Determine the domain of the function.
- Locate the – and -intercepts.
- Evaluate and to determine the end behavior. If either of these limits is a finite number then is a horizontal asymptote. If either of these limits is or determine whether has an oblique asymptote. If is a rational function such that where the degree of the numerator is greater than the degree of the denominator, then can be written as
where the degree of is less than the degree of The values of approach the values of as If is a linear function, it is known as an

*oblique asymptote*. - Determine whether has any vertical asymptotes.
- Calculate Find all critical points and determine the intervals where is increasing and where is decreasing. Determine whether has any local extrema.
- Calculate Determine the intervals where is concave up and where is concave down. Use this information to determine whether has any inflection points. The second derivative can also be used as an alternate means to determine or verify that has a local extremum at a critical point.

Now let’s use this strategy to graph several different functions. We start by graphing a polynomial function.

### Sketching a Graph of a Polynomial

Sketch a graph of

#### Solution

Step 1. Since is a polynomial, the domain is the set of all real numbers.

Step 2. When Therefore, the -intercept is To find the -intercepts, we need to solve the equation gives us the -intercepts and

Step 3. We need to evaluate the end behavior of As and Therefore, As and Therefore, To get even more information about the end behavior of we can multiply the factors of When doing so, we see that

Since the leading term of is we conclude that behaves like as

Step 4. Since is a polynomial function, it does not have any vertical asymptotes.

Step 5. The first derivative of is

Therefore, has two critical points: Divide the interval into the three smaller intervals: and Then, choose test points and from these intervals and evaluate the sign of at each of these test points, as shown in the following table.

Interval | Test Point | Sign of Derivative | Conclusion |
---|---|---|---|

is increasing. | |||

is decreasing. | |||

is increasing. |

From the table, we see that has a local maximum at and a local minimum at Evaluating at those two points, we find that the local maximum value is and the local minimum value is

Step 6. The second derivative of is

The second derivative is zero at Therefore, to determine the concavity of divide the interval into the smaller intervals and and choose test points and to determine the concavity of on each of these smaller intervals as shown in the following table.

Interval | Test Point | Sign of | Conclusion |
---|---|---|---|

is concave down. | |||

is concave up. |

We note that the information in the preceding table confirms the fact, found in step 5, that has a local maximum at and a local minimum at In addition, the information found in step 5—namely, has a local maximum at and a local minimum at and at those points—combined with the fact that changes sign only at confirms the results found in step 6 on the concavity of

Combining this information, we arrive at the graph of shown in the following graph.

Sketch a graph of

#### Solution

#### Hint

is a fourth-degree polynomial.

### Sketching a Rational Function

Sketch the graph of

#### Solution

Step 1. The function is defined as long as the denominator is not zero. Therefore, the domain is the set of all real numbers except

Step 2. Find the intercepts. If then so 0 is an intercept. If then which implies Therefore, is the only intercept.

Step 3. Evaluate the limits at infinity. Since is a rational function, divide the numerator and denominator by the highest power in the denominator: We obtain

Therefore, has a horizontal asymptote of as and

Step 4. To determine whether has any vertical asymptotes, first check to see whether the denominator has any zeroes. We find the denominator is zero when To determine whether the lines or are vertical asymptotes of evaluate and By looking at each one-sided limit as we see that

In addition, by looking at each one-sided limit as we find that

Step 5. Calculate the first derivative:

Critical points occur at points where or is undefined. We see that when The derivative is not undefined at any point in the domain of However, are not in the domain of Therefore, to determine where is increasing and where is decreasing, divide the interval into four smaller intervals: and and choose a test point in each interval to determine the sign of in each of these intervals. The values and are good choices for test points as shown in the following table.

Interval | Test Point | Sign of | Conclusion |
---|---|---|---|

is decreasing. | |||

is decreasing. | |||

is increasing. | |||

is increasing. |

From this analysis, we conclude that has a local minimum at but no local maximum.

Step 6. Calculate the second derivative:

To determine the intervals where is concave up and where is concave down, we first need to find all points where or is undefined. Since the numerator for any is never zero. Furthermore, is not undefined for any in the domain of However, as discussed earlier, are not in the domain of Therefore, to determine the concavity of we divide the interval into the three smaller intervals and and choose a test point in each of these intervals to evaluate the sign of in each of these intervals. The values and are possible test points as shown in the following table.

Interval | Test Point | Sign of | Conclusion |
---|---|---|---|

is concave down. | |||

is concave up. | |||

is concave down. |

Combining all this information, we arrive at the graph of shown below. Note that, although changes concavity at and there are no inflection points at either of these places because is not continuous at or

Sketch a graph of

#### Solution

#### Hint

A line is a horizontal asymptote of if the limit as or the limit as of is A line is a vertical asymptote if at least one of the one-sided limits of as is or

### Sketching a Rational Function with an Oblique Asymptote

Sketch the graph of

#### Solution

Step 1. The domain of is the set of all real numbers except

Step 2. Find the intercepts. We can see that when so is the only intercept.

Step 3. Evaluate the limits at infinity. Since the degree of the numerator is one more than the degree of the denominator, must have an oblique asymptote. To find the oblique asymptote, use long division of polynomials to write

Since as approaches the line as The line is an oblique asymptote for

Step 4. To check for vertical asymptotes, look at where the denominator is zero. Here the denominator is zero at Looking at both one-sided limits as we find

Therefore, is a vertical asymptote, and we have determined the behavior of as approaches 1 from the right and the left.

Step 5. Calculate the first derivative:

We have when Therefore, and are critical points. Since is undefined at we need to divide the interval into the smaller intervals and and choose a test point from each interval to evaluate the sign of in each of these smaller intervals. For example, let and be the test points as shown in the following table.

Interval | Test Point | Sign of | Conclusion |
---|---|---|---|

is increasing. | |||

is decreasing. | |||

is decreasing. | |||

is increasing. |

From this table, we see that has a local maximum at and a local minimum at The value of at the local maximum is and the value of at the local minimum is Therefore, and are important points on the graph.

Step 6. Calculate the second derivative:

We see that is never zero or undefined for in the domain of Since is undefined at to check concavity we just divide the interval into the two smaller intervals and and choose a test point from each interval to evaluate the sign of in each of these intervals. The values and are possible test points as shown in the following table.

Interval | Test Point | Sign of | Conclusion |
---|---|---|---|

is concave down. | |||

is concave up. |

From the information gathered, we arrive at the following graph for

Find the oblique asymptote for

#### Solution

#### Hint

Use long division of polynomials.

### Sketching the Graph of a Function with a Cusp

Sketch a graph of

#### Solution

Step 1. Since the cube-root function is defined for all real numbers and the domain of is all real numbers.

Step 2: To find the -intercept, evaluate Since the -intercept is To find the -intercept, solve The solution of this equation is so the -intercept is

Step 3: Since the function continues to grow without bound as and

Step 4: The function has no vertical asymptotes.

Step 5: To determine where is increasing or decreasing, calculate We find

This function is not zero anywhere, but it is undefined when Therefore, the only critical point is Divide the interval into the smaller intervals and and choose test points in each of these intervals to determine the sign of in each of these smaller intervals. Let and be the test points as shown in the following table.

Interval | Test Point | Sign of | Conclusion |
---|---|---|---|

is decreasing. | |||

is increasing. |

We conclude that has a local minimum at Evaluating at we find that the value of at the local minimum is zero. Note that is undefined, so to determine the behavior of the function at this critical point, we need to examine Looking at the one-sided limits, we have

Therefore, has a cusp at

Step 6: To determine concavity, we calculate the second derivative of

We find that is defined for all but is undefined when Therefore, divide the interval into the smaller intervals and and choose test points to evaluate the sign of in each of these intervals. As we did earlier, let and be test points as shown in the following table.

Interval | Test Point | Sign of | Conclusion |
---|---|---|---|

is concave down. | |||

is concave down. |

From this table, we conclude that is concave down everywhere. Combining all of this information, we arrive at the following graph for

Consider the function Determine the point on the graph where a cusp is located. Determine the end behavior of

#### Solution

The function has a cusp at For end behavior,

#### Hint

A function has a cusp at a point if exists, is undefined, one of the one-sided limits as of is and the other one-sided limit is

### Key Concepts

- The limit of is as (or as if the values become arbitrarily close to as becomes sufficiently large.
- The limit of is as if becomes arbitrarily large as becomes sufficiently large. The limit of is as if and becomes arbitrarily large as becomes sufficiently large. We can define the limit of as approaches similarly.
- For a polynomial function where the end behavior is determined by the leading term If approaches or at each end.
- For a rational function the end behavior is determined by the relationship between the degree of and the degree of If the degree of is less than the degree of the line is a horizontal asymptote for If the degree of is equal to the degree of then the line is a horizontal asymptote, where and are the leading coefficients of and respectively. If the degree of is greater than the degree of then approaches or at each end.

For the following exercises, examine the graphs. Identify where the vertical asymptotes are located.

**1.**

#### Solution

**2.**

**3.**

#### Solution

**4.**

**5.**

#### Solution

For the following functions determine whether there is an asymptote at Justify your answer without graphing on a calculator.

**6.**

**7.**

#### Solution

Yes, there is a vertical asymptote

**8.**

**9.**

#### Solution

Yes, there is vertical asymptote

**10.**

For the following exercises, evaluate the limit.

**11.**

#### Solution

0

**12.**

**13.**

#### Solution

**14.**

**15.**

#### Solution

**16.**

**17.**

#### Solution

-2

**18.**

**19.**

#### Solution

-4

**20.**

For the following exercises, find the horizontal and vertical asymptotes.

**21.**

#### Solution

Horizontal: none, vertical:

**22.**

**23.**

#### Solution

Horizontal: none, vertical:

**24.**

**25.**

#### Solution

Horizontal: none, vertical: none

**26.**

**27.**

#### Solution

Horizontal: vertical:

**28.**

#### Solution

Horizontal: vertical: and

**29.**

**30.**

#### Solution

Horizontal: vertical:

**31.**

**32.**

#### Solution

Horizontal: none, vertical: none

**33.**

For the following exercises, construct a function that has the given asymptotes.

**34.** and

#### Solution

Answers will vary, for example:

**35.** and

**36.** [latex]x=-1[/latex]

#### Solution

Answers will vary, for example:

**37.**

For the following exercises, graph the function on a graphing calculator on the window and estimate the horizontal asymptote or limit. Then, calculate the actual horizontal asymptote or limit.

**38. [T]**

#### Solution

**39. [T]**

**40. [T]**

#### Solution

**41. [T]**

**42. [T]**

#### Solution

For the following exercises, draw a graph of the functions without using a calculator. Be sure to notice all important features of the graph: local maxima and minima, inflection points, and asymptotic behavior.

**43.**

**44.**

#### Solution

**45.**

**46.**

#### Solution

**47.**

**48.**

#### Solution

**49.**

**50.** on

#### Solution

**51.**

**52.**

#### Solution

**53.**

**54.**

#### Solution

**55.** For to have an asymptote at then the polynomials and must have what relation?

**56.** For to have an asymptote at then the polynomials and must have what relation?

#### Solution

must have have as a factor, where has as a factor.

**57.** If has asymptotes at and then has what asymptotes?

**58.** Both and have asymptotes at and What is the most obvious difference between these two functions?

#### Solution

**59.** True or false: Every ratio of polynomials has vertical asymptotes.

## Glossary

- end behavior
- the behavior of a function as and

- horizontal asymptote
- if or then is a horizontal asymptote of

- infinite limit at infinity
- a function that becomes arbitrarily large as becomes large

- limit at infinity
- the limiting value, if it exists, of a function as or

- oblique asymptote
- the line if approaches it as or

## Hint