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8

Learning Objectives

  • Calculate the limit of a function as x increases or decreases without bound.
  • Recognize a horizontal asymptote on the graph of a function.
  • Estimate the end behavior of a function as x increases or decreases without bound.
  • Recognize an oblique asymptote on the graph of a function.

To graph a function f defined on an unbounded domain, we also need to know the behavior of f as x\to \pm \infty . In this section, we define limits at infinity and show how these limits affect the graph of a 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 \underset{x\to a}{\text{lim}}f(x)=L means f(x) becomes arbitrarily close to L as long as x is sufficiently close to a. We can extend this idea to limits at infinity. For example, consider the function f(x)=2+\frac{1}{x}. As can be seen graphically in (Figure) and numerically in (Figure), as the values of x get larger, the values of f(x) approach 2. We say the limit as x approaches \infty of f(x) is 2 and write \underset{x\to \infty }{\text{lim}}f(x)=2. Similarly, for x<0, as the values |x| get larger, the values of f(x) approaches 2. We say the limit as x approaches -\infty of f(x) is 2 and write \underset{x\to a}{\text{lim}}f(x)=2.

The function f(x) 2 + 1/x is graphed. The function starts negative near y = 2 but then decreases to −∞ near x = 0. The function then decreases from ∞ near x = 0 and gets nearer to y = 2 as x increases. There is a horizontal line denoting the asymptote y = 2.
Figure 1. The function approaches the asymptote y=2 as x approaches \pm \infty .
Values of a function f as x\to \pm \infty
x 10 100 1,000 10,000
2+\frac{1}{x} 2.1 2.01 2.001 2.0001
x -10 -100 -1000 -10,000
2+\frac{1}{x} 1.9 1.99 1.999 1.9999

More generally, for any function f, we say the limit as x\to \infty of f(x) is L if f(x) becomes arbitrarily close to L as long as x is sufficiently large. In that case, we write \underset{x\to a}{\text{lim}}f(x)=L. Similarly, we say the limit as x\to -\infty of f(x) is L if f(x) becomes arbitrarily close to L as long as x<0 and |x| is sufficiently large. In that case, we write \underset{x\to -\infty }{\text{lim}}f(x)=L. We now look at the definition of a function having a limit at infinity.

Definition

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

\underset{x\to \infty }{\text{lim}}f(x)=L.

If the values of f(x) becomes arbitrarily close to L for x<0 as |x| becomes sufficiently large, we say that the function f has a limit at negative infinity and write

\underset{x\to \infty }{\text{lim}}f(x)=L.

If the values f(x) are getting arbitrarily close to some finite value L as x\to \infty or x\to -\infty , the graph of f approaches the line y=L. In that case, the line y=L is a horizontal asymptote of f ((Figure)). For example, for the function f(x)=\frac{1}{x}, since \underset{x\to \infty }{\text{lim}}f(x)=0, the line y=0 is a horizontal asymptote of f(x)=\frac{1}{x}.

Definition

If \underset{x\to \infty }{\text{lim}}f(x)=L or \underset{x\to -\infty }{\text{lim}}f(x)=L, we say the line y=L is a horizontal asymptote of f.

The figure is broken up into two figures labeled a and b. Figure a shows a function f(x) approaching but never touching a horizontal dashed line labeled L from above. Figure b shows a function f(x) approaching but never a horizontal dashed line labeled M from below.
Figure 2. (a) As x\to \infty , the values of f are getting arbitrarily close to L. The line y=L is a horizontal asymptote of f. (b) As x\to -\infty , the values of f are getting arbitrarily close to M. The line y=M is a horizontal asymptote of f.

A function cannot cross a vertical asymptote because the graph must approach infinity (or -\infty ) from at least one direction as x 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 f(x)=\frac{( \cos x)}{x}+1 shown in (Figure) intersects the horizontal asymptote y=1 an infinite number of times as it oscillates around the asymptote with ever-decreasing amplitude.

The function f(x) = (cos x)/x + 1 is shown. It decreases from (0, ∞) and then proceeds to oscillate around y = 1 with decreasing amplitude.
Figure 3. The graph of f(x)=( \cos x)\text{/}x+1 crosses its horizontal asymptote y=1 an infinite number of times.

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 f, evaluate \underset{x\to \infty }{\text{lim}}f(x) and \underset{x\to -\infty }{\text{lim}}f(x). Determine the horizontal asymptote(s) for f.

  1. f(x)=5-\frac{2}{{x}^{2}}
  2. f(x)=\frac{ \sin x}{x}
  3. f(x)={ \tan }^{-1}(x)

Solution

  1. Using the algebraic limit laws, we have \underset{x\to \infty }{\text{lim}}(5-\frac{2}{{x}^{2}})=\underset{x\to \infty }{\text{lim}}5-2(\underset{x\to \infty }{\text{lim}}\frac{1}{x}).(\underset{x\to \infty }{\text{lim}}\frac{1}{x})=5-2\cdot 0=5.Similarly, \underset{x\to \infty }{\text{lim}}f(x)=5.Therefore, f(x)=5-\frac{2}{{x}^{2}} has a horizontal asymptote of y=5 and f approaches this horizontal asymptote as x\to \pm \infty as shown in the following graph.
    The function f(x) = 5 – 2/x2 is graphed. The function approaches the horizontal asymptote y = 5 as x approaches ±∞.
    Figure 4. This function approaches a horizontal asymptote as x\to \pm \infty .
  2. Since -1\le \sin x\le 1 for all x, we have
    \frac{-1}{x}\le \frac{ \sin x}{x}\le \frac{1}{x}

    for all x\ne 0. Also, since

    \underset{x\to \infty }{\text{lim}}\frac{-1}{x}=0=\underset{x\to \infty }{\text{lim}}\frac{1}{x},

    we can apply the squeeze theorem to conclude that

    \underset{x\to \infty }{\text{lim}}\frac{ \sin x}{x}=0.

    Similarly,

    \underset{x\to -\infty }{\text{lim}}\frac{ \sin x}{x}=0.

     

    Thus, f(x)=\frac{ \sin x}{x} has a horizontal asymptote of y=0 and f(x) approaches this horizontal asymptote as x\to \pm \infty as shown in the following graph.

    The function f(x) = (sin x)/x is shown. It has a global maximum at (0, 1) and then proceeds to oscillate around y = 0 with decreasing amplitude.
    Figure 5. This function crosses its horizontal asymptote multiple times.
  3. To evaluate \underset{x\to \infty }{\text{lim}}{ \tan }^{-1}(x) and \underset{x\to -\infty }{\text{lim}}{ \tan }^{-1}(x), we first consider the graph of y= \tan (x) over the interval (-\pi \text{/}2,\pi \text{/}2) as shown in the following graph.
    The function f(x) = tan x is shown. It increases from (−π/2, −∞), passes through the origin, and then increases toward (π/2, ∞). There are vertical dashed lines marking x = ±π/2.
    The graph of \tan x has vertical asymptotes at x=\pm \frac{\pi }{2}

Since

\underset{x\to {(\pi \text{/}2)}^{-}}{\text{lim}} \tan x=\infty ,

it follows that

\underset{x\to \infty }{\text{lim}}{ \tan }^{-1}(x)=\frac{\pi }{2}.

Similarly, since

\underset{x\to {(\pi \text{/}2)}^{+}}{\text{lim}} \tan x=-\infty ,

it follows that

\underset{x\to -\infty }{\text{lim}}{ \tan }^{-1}(x)=-\frac{\pi }{2}.

As a result, y=\frac{\pi }{2} and y=-\frac{\pi }{2} are horizontal asymptotes of f(x)={ \tan }^{-1}(x) as shown in the following graph.

The function f(x) = tan−1 x is shown. It increases from (−∞, −π/2), passes through the origin, and then increases toward (∞, π/2). There are horizontal dashed lines marking y = ±π/2.
Figure 7. This function has two horizontal asymptotes.

Evaluate \underset{x\to -\infty }{\text{lim}}(3+\frac{4}{x}) and \underset{x\to \infty }{\text{lim}}(3+\frac{4}{x}). Determine the horizontal asymptotes of f(x)=3+\frac{4}{x}, if any.

Hint

\underset{x\to \pm \infty }{\text{lim}}1\text{/}x=0

Solution

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

Infinite Limits at Infinity

Sometimes the values of a function f become arbitrarily large as x\to \infty (or as x\to -\infty ). In this case, we write \underset{x\to \infty }{\text{lim}}f(x)=\infty (or \underset{x\to -\infty }{\text{lim}}f(x)=\infty ). On the other hand, if the values of f are negative but become arbitrarily large in magnitude as x\to \infty (or as x\to -\infty ), we write \underset{x\to \infty }{\text{lim}}f(x)=-\infty (or \underset{x\to -\infty }{\text{lim}}f(x)=-\infty ).

For example, consider the function f(x)={x}^{3}. As seen in (Figure) and (Figure), as x\to \infty the values f(x) become arbitrarily large. Therefore, \underset{x\to \infty }{\text{lim}}{x}^{3}=\infty . On the other hand, as x\to -\infty , the values of f(x)={x}^{3} are negative but become arbitrarily large in magnitude. Consequently, \underset{x\to -\infty }{\text{lim}}{x}^{3}=-\infty .

Values of a power function as x\to \pm \infty
x 10 20 50 100 1000
{x}^{3} 1000 8000 125,000 1,000,000 1,000,000,000
x -10 -20 -50 -100 -1000
{x}^{3} -1000 -8000 -125,000 -1,000,000 -1,000,000,000
The function f(x) = x3 is graphed. It is apparent that this function rapidly approaches infinity as x approaches infinity.
Figure 8. For this function, the functional values approach infinity as x\to \pm \infty .

Definition

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

\underset{x\to \infty }{\text{lim}}f(x)=\infty .

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

\underset{x\to \infty }{\text{lim}}f(x)=-\infty .

if f(x)<0 and |f(x)| becomes arbitrarily large for x sufficiently large. Similarly, we can define infinite limits as x\to -\infty .

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 f has a limit at infinity, if there exists a real number L such that for all \epsilon \symbol{"3E} 0, there exists N \symbol{"3E} 0 such that

|f(x)-L|<\epsilon

for all x \symbol{"3E} N. In that case, we write

\underset{x\to \infty }{\text{lim}}f(x)=L

(see (Figure)).

We say a function f has a limit at negative infinity if there exists a real number L such that for all \epsilon \symbol{"3E} 0, there exists N<0 such that

|f(x)-L|<\epsilon

for all x<N. In that case, we write

\underset{x\to -\infty }{\text{lim}}f(x)=L.
The function f(x) is graphed, and it has a horizontal asymptote at L. L is marked on the y axis, as is L + and L – On the x axis, N is marked as the value of x such that f(x) = L + .
Figure 9. For a function with a limit at infinity, for all x \symbol{"3E} N, |f(x)-L|<\epsilon .

Earlier in this section, we used graphical evidence in (Figure) and numerical evidence in (Figure) to conclude that \underset{x\to \infty }{\text{lim}}(\frac{2+1}{x})=2. 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 \underset{x\to \infty }{\text{lim}}(\frac{2+1}{x})=2.

Solution

Let \epsilon \symbol{"3E} 0. Let N=\frac{1}{\epsilon }. Therefore, for all x \symbol{"3E} N, we have

|2+\frac{1}{x}-2|=|\frac{1}{x}|=\frac{1}{x}<\frac{1}{N}=\epsilon \text{.}

Use the formal definition of limit at infinity to prove that \underset{x\to \infty }{\text{lim}}(\frac{3-1}{{x}^{2}})=3.

Hint

Let N=\frac{1}{\sqrt{\epsilon }}.

Solution

Let \epsilon \symbol{"3E} 0. Let N=\frac{1}{\sqrt{\epsilon }}. Therefore, for all x \symbol{"3E} N, we have

|3-\frac{1}{{x}^{2}}-3|=\frac{1}{{x}^{2}}<\frac{1}{{N}^{2}}=\epsilon

Therefore, \underset{x\to \infty }{\text{lim}}(3-1\text{/}{x}^{2})=3.

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

Definition

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

\underset{x\to \infty }{\text{lim}}f(x)=\infty

if for all M \symbol{"3E} 0, there exists an N \symbol{"3E} 0 such that

f(x) \symbol{"3E} M

for all x \symbol{"3E} N (see (Figure)).

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

\underset{x\to \infty }{\text{lim}}f(x)=-\infty

if for all M<0, there exists an N \symbol{"3E} 0 such that

f(x)<M

for all x \symbol{"3E} N.

Similarly we can define limits as x\to -\infty .

The function f(x) is graphed. It continues to increase rapidly after x = N, and f(N) = M.
Figure 10. For a function with an infinite limit at infinity, for all x \symbol{"3E} N, f(x) \symbol{"3E} M.

Earlier, we used graphical evidence ((Figure)) and numerical evidence ((Figure)) to conclude that \underset{x\to \infty }{\text{lim}}{x}^{3}=\infty . 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 \underset{x\to \infty }{\text{lim}}{x}^{3}=\infty .

Solution

Let M \symbol{"3E} 0. Let N=\sqrt[3]{M}. Then, for all x \symbol{"3E} N, we have

{x}^{3} \symbol{"3E} {N}^{3}={(\sqrt[3]{M})}^{3}=M.

Therefore, \underset{x\to \infty }{\text{lim}}{x}^{3}=\infty .

Use the formal definition of infinite limit at infinity to prove that \underset{x\to \infty }{\text{lim}}3{x}^{2}=\infty .

Hint

Let N=\sqrt{\frac{M}{3}}.

Solution

Let M \symbol{"3E} 0. Let N=\sqrt{\frac{M}{3}}. Then, for all x \symbol{"3E} N, we have

3{x}^{2} \symbol{"3E} 3{N}^{2}=3{(\sqrt{\frac{M}{3}})}^{2}{2}^{}=\frac{3M}{3}=M

End Behavior

The behavior of a function as x\to \pm\infty 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:

  1. The function f(x) approaches a horizontal asymptote y=L.
  2. The function f(x)\to \infty or f(x)\to -\infty .
  3. The function does not approach a finite limit, nor does it approach \infty or -\infty . 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 f(x)={x}^{n} where n is a positive integer. From (Figure) and (Figure), we see that

\underset{x\to \infty }{\text{lim}}{x}^{n}=\infty ;n=1,2,3\text{,…}

and

\underset{x\to -\infty }{\text{lim}}{x}^{n}=\bigg\{\begin{array}{c}\infty ;n=2,4,6\text{,…}\hfill \\ -\infty ;n=1,3,5\text{,…}\hfill \end{array}.
The functions x2, x4, and x6 are graphed, and it is apparent that as the exponent grows the functions increase more quickly.
Figure 11. For power functions with an even power of n, \underset{x\to \infty }{\text{lim}}{x}^{n}=\infty =\underset{x\to -\infty }{\text{lim}}{x}^{n}.
The functions x, x3, and x5 are graphed, and it is apparent that as the exponent grows the functions increase more quickly.
Figure 12. For power functions with an odd power of n, \underset{x\to \infty }{\text{lim}}{x}^{n}=\infty and \underset{x\to -\infty }{\text{lim}}{x}^{n}=-\infty .

Using these facts, it is not difficult to evaluate \underset{x\to \infty }{\text{lim}}c{x}^{n} and \underset{x\to -\infty }{\text{lim}}c{x}^{n}, where c is any constant and n is a positive integer. If c \symbol{"3E} 0, the graph of y=c{x}^{n} is a vertical stretch or compression of y={x}^{n}, and therefore

\underset{x\to \infty }{\text{lim}}c{x}^{n}=\underset{x\to \infty }{\text{lim}}{x}^{n}\text{ and }\underset{x\to -\infty }{\text{lim}}c{x}^{n}=\underset{x\to -\infty }{\text{lim}}{x}^{n}\text{ if }c \symbol{"3E} 0.

If c<0, the graph of y=c{x}^{n} is a vertical stretch or compression combined with a reflection about the x-axis, and therefore

\underset{x\to \infty }{\text{lim}}c{x}^{n}=-\underset{x\to \infty }{\text{lim}}{x}^{n}\text{ and }\underset{x\to -\infty }{\text{lim}}c{x}^{n}=-\underset{x\to -\infty }{\text{lim}}{x}^{n}\text{ if }c<0.

If c=0,y=c{x}^{n}=0, in which case \underset{x\to \infty }{\text{lim}}c{x}^{n}=0=\underset{x\to -\infty }{\text{lim}}c{x}^{n}.

Limits at Infinity for Power Functions

For each function f, evaluate \underset{x\to \infty }{\text{lim}}f(x) and \underset{x\to -\infty }{\text{lim}}f(x).

  1. f(x)=-5{x}^{3}
  2. f(x)=2{x}^{4}
  3. f(x)=2{x}^{3}-{x}^{2}

Solution

  1. Since the coefficient of {x}^{3} is -5, the graph of f(x)=-5{x}^{3} involves a vertical stretch and reflection of the graph of y={x}^{3} about the x-axis. Therefore, \underset{x\to \infty }{\text{lim}}(-5{x}^{3})=-\infty and \underset{x\to -\infty }{\text{lim}}(-5{x}^{3})=\infty .
  2. Since the coefficient of {x}^{4} is 2, the graph of f(x)=2{x}^{4} is a vertical stretch of the graph of y={x}^{4}. Therefore, \underset{x\to \infty }{\text{lim}}2{x}^{4}=\infty and \underset{x\to -\infty }{\text{lim}}2{x}^{4}=\infty .
  3. This is an \infty - \infty form so limit laws don’t help. However, by factoring we get
    \underset{x\to \pm\infty }{\text{lim}}\left( 2x^3-x^2 \right) = \underset{x\to \pm\infty }{\text{lim}}x^3\left( 2-\frac{1}{x} \right).As x\to\infty, the power x^3 \to \infty where the second factor gets close to 2. Hence the limit is \infty.As x\to -\infty, the power x^3 \to -\infty where the second factor gets close to 2. Hence the limit is -\infty.

Let f(x)=-3{x}^{4}+4x^3. Find \underset{x\to \infty }{\text{lim}}f(x).

Hint

The coefficient -3 is negative.

Solution

-\infty

End Behavior for Algebraic Functions

The end behavior for rational functions and functions involving radicals is a little more complicated than for polynomials. To evaluate the limits at infinity for a rational function, we divide the numerator and denominator by the highest power of x appearing in the denominator. This determines which term in the overall expression dominates the behavior of the function at large values of x.

Determining End Behavior for Rational Functions

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

  1. f(x)=\frac{3x-1}{2x+5}
  2. f(x)=\frac{3{x}^{2}+2x}{4{x}^{3}-5x+7}
  3. f(x)=\frac{3{x}^{2}+4x}{x+2}

Solution

  1. The highest power of x in the denominator is x. Therefore, dividing the numerator and denominator by x and applying the algebraic limit laws, we see that
    \begin{array}{cc}\hfill \underset{x\to \pm \infty }{\text{lim}}\frac{3x-1}{2x+5}& =\underset{x\to \pm \infty }{\text{lim}}\frac{3-1\text{/}x}{2+5\text{/}x}\hfill \\ & =\frac{\underset{x\to \pm \infty }{\text{lim}}(3-1\text{/}x)}{\underset{x\to \pm \infty }{\text{lim}}(2+5\text{/}x)}\hfill \\ & =\frac{\underset{x\to \pm \infty }{\text{lim}}3-\underset{x\to \pm \infty }{\text{lim}}1\text{/}x}{\underset{x\to \pm \infty }{\text{lim}}2+\underset{x\to \pm \infty }{\text{lim}}5\text{/}x}\hfill \\ & =\frac{3-0}{2+0}=\frac{3}{2}.\hfill \end{array}

    Since \underset{x\to \pm \infty }{\text{lim}}f(x)=\frac{3}{2}, we know that y=\frac{3}{2} is a horizontal asymptote for this function as shown in the following graph.

    The function f(x) = (3x + 1)/(2x + 5) is plotted as is its horizontal asymptote at y = 3/2.
    Figure 14. The graph of this rational function approaches a horizontal asymptote as x\to \pm \infty .
  2. Since the largest power of x appearing in the denominator is {x}^{3}, divide the numerator and denominator by {x}^{3}. After doing so and applying algebraic limit laws, we obtain
    \underset{x\to \pm \infty }{\text{lim}}\frac{3{x}^{2}+2x}{4{x}^{3}-5x+7}=\underset{x\to \pm \infty }{\text{lim}}\frac{3\text{/}x+2\text{/}{x}^{2}}{4-5\text{/}{x}^{2}+7\text{/}{x}^{3}}=\frac{3.0+2.0}{4-5.0+7.0}=0.

    Therefore f has a horizontal asymptote of y=0 as shown in the following graph.

    The function f(x) = (3x2 + 2x)/(4x2 – 5x + 7) is plotted as is its horizontal asymptote at y = 0.
    Figure 15. The graph of this rational function approaches the horizontal asymptote y=0 as x\to \pm \infty .
  3. Dividing the numerator and denominator by x, we have
    \underset{x\to \pm \infty }{\text{lim}}\frac{3{x}^{2}+4x}{x+2}=\underset{x\to \pm \infty }{\text{lim}}\frac{3x+4}{1+2\text{/}x}.

    As x\to \pm \infty , the denominator approaches 1. As x\to \infty , the numerator approaches +\infty . As x\to -\infty , the numerator approaches -\infty . Therefore \underset{x\to \infty }{\text{lim}}f(x)=\infty , whereas \underset{x\to -\infty }{\text{lim}}f(x)=-\infty as shown in the following figure.

    The function f(x) = (3x2 + 4x)/(x + 2) is plotted. It appears to have a diagonal asymptote as well as a vertical asymptote at x = −2.
    Figure 16. As x\to \infty , the values f(x)\to \infty . As x\to -\infty , the values f(x)\to -\infty .

Evaluate \underset{x\to \pm \infty }{\text{lim}}\frac{3{x}^{2}+2x-1}{5{x}^{2}-4x+7} and use these limits to determine the end behavior of f(x)=\frac{3{x}^{2}+2x-2}{5{x}^{2}-4x+7}.

Hint

Divide the numerator and denominator by {x}^{2}.

Solution

\frac{3}{5}

Before proceeding, consider the graph of f(x)=\frac{(3{x}^{2}+4x)}{(x+2)} shown in (Figure). As x\to \infty and x\to -\infty , the graph of f appears almost linear. Although f is certainly not a linear function, we now investigate why the graph of f seems to be approaching a linear function. First, using long division of polynomials, we can write

f(x)=\frac{3{x}^{2}+4x}{x+2}=3x-2+\frac{4}{x+2}.

Since \frac{4}{(x+2)}\to 0 as x\to \pm \infty , we conclude that

\underset{x\to \pm \infty }{\text{lim}}(f(x)-(3x-2))=\underset{x\to \pm \infty }{\text{lim}}\frac{4}{x+2}=0.

Therefore, the graph of f approaches the line y=3x-2 as x\to \pm \infty . This line is known as an oblique asymptote for f ((Figure)).

The function f(x) = (3x2 + 4x)/(x + 2) is plotted as is its diagonal asymptote y = 3x – 2.
Figure 17. The graph of the rational function f(x)=(3{x}^{2}+4x)\text{/}(x+2) approaches the oblique asymptote y=3x-2\text{as}x\to \pm \infty .

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 x\to \infty and x\to -\infty for f(x)=\frac{3x-2}{\sqrt{4{x}^{2}+5}} and describe the end behavior of f.

Solution

Let’s use the same strategy as we did for rational functions: divide the numerator and denominator by a power of x. To determine the appropriate power of x, consider the expression \sqrt{4{x}^{2}+5} in the denominator. Since

\sqrt{4{x}^{2}+5}\approx \sqrt{4{x}^{2}}=2|x|

for large values of x in effect x appears just to the first power in the denominator. Therefore, we divide the numerator and denominator by |x|. Then, using the fact that |x|=x for x \symbol{"3E} 0, |x|=-x for x<0, and |x|=\sqrt{{x}^{2}} for all x, we calculate the limits as follows:

\begin{array}{ccc}\hfill \underset{x\to \infty }{\text{lim}}\frac{3x-2}{\sqrt{4{x}^{2}+5}}& =\hfill & \underset{x\to \infty }{\text{lim}}\frac{(1\text{/}|x|)(3x-2)}{(1\text{/}|x|)\sqrt{4{x}^{2}+5}}\hfill \\ & =\hfill & \underset{x\to \infty }{\text{lim}}\frac{(1\text{/}x)(3x-2)}{\sqrt{(1\text{/}{x}^{2})(4{x}^{2}+5)}}\hfill \\ & =\hfill & \underset{x\to \infty }{\text{lim}}\frac{3-2\text{/}x}{\sqrt{4+5\text{/}{x}^{2}}}=\frac{3}{\sqrt{4}}=\frac{3}{2}\hfill \\ \hfill \underset{x\to -\infty }{\text{lim}}\frac{3x-2}{\sqrt{4{x}^{2}+5}}& =\hfill & \underset{x\to -\infty }{\text{lim}}\frac{(1\text{/}|x|)(3x-2)}{(1\text{/}|x|)\sqrt{4{x}^{2}+5}}\hfill \\ & =\hfill & \underset{x\to -\infty }{\text{lim}}\frac{(-1\text{/}x)(3x-2)}{\sqrt{(1\text{/}{x}^{2})(4{x}^{2}+5)}}\hfill \\ & =\hfill & \underset{x\to -\infty }{\text{lim}}\frac{-3+2\text{/}x}{\sqrt{4+5\text{/}{x}^{2}}}=\frac{-3}{\sqrt{4}}=\frac{-3}{2}.\hfill \end{array}

Therefore, f(x) approaches the horizontal asymptote y=\frac{3}{2} as x\to \infty and the horizontal asymptote y=-\frac{3}{2} as x\to -\infty as shown in the following graph.

The function f(x) = (3x − 2)/(the square root of the quantity (4x2 + 5)) is plotted. It has two horizontal asymptotes at y = ±3/2, and it crosses y = −3/2 before converging toward it from below.
Figure 18. This function has two horizontal asymptotes and it crosses one of the asymptotes.

Evaluate \underset{x\to \infty }{\text{lim}}\frac{\sqrt{3{x}^{2}+4}}{x+6}.

Hint

Divide the numerator and denominator by |x|.

Solution

\pm \sqrt{3}

Determining End Behavior for Transcendental Functions

The six basic trigonometric functions are periodic and do not approach a finite limit as x\to \pm \infty . For example, \sin x oscillates between 1\text{ and }-1 ((Figure)). The tangent function x has an infinite number of vertical asymptotes as x\to \pm \infty ; therefore, it does not approach a finite limit nor does it approach \pm \infty as x\to \pm \infty as shown in (Figure).

The function f(x) = sin x is graphed.
Figure 19. The function f(x)= \sin x oscillates between 1\text{ and }-1 as x\to \pm \infty
The function f(x) = tan x is graphed.
Figure 20. The function f(x)= \tan x does not approach a limit and does not approach \pm \infty as x\to \pm \infty

Recall that for any base b \symbol{"3E} 0,b\ne 1, the function y={b}^{x} is an exponential function with domain (-\infty ,\infty ) and range (0,\infty ). If b \symbol{"3E} 1,y={b}^{x} is increasing over `(-\infty ,\infty ). If 0<b<1, y={b}^{x} is decreasing over (-\infty ,\infty ). For the natural exponential function f(x)={e}^{x}, e\approx 2.718 \symbol{"3E} 1. Therefore, f(x)={e}^{x} is increasing on `(-\infty ,\infty ) and the range is `(0,\infty ). The exponential function f(x)={e}^{x} approaches \infty as x\to \infty and approaches 0 as x\to -\infty as shown in (Figure) and (Figure).

End behavior of the natural exponential function
x -5 -2 0 2 5
{e}^{x} 0.00674 0.135 1 7.389 148.413
The function f(x) = ex is graphed.
Figure 21. The exponential function approaches zero as x\to -\infty and approaches \infty as x\to \infty .

Recall that the natural logarithm function f(x)=\text{ln}(x) is the inverse of the natural exponential function y={e}^{x}. Therefore, the domain of f(x)=\text{ln}(x) is (0,\infty ) and the range is (-\infty ,\infty ). The graph of f(x)=\text{ln}(x) is the reflection of the graph of y={e}^{x} about the line y=x. Therefore, \text{ln}(x)\to -\infty as x\to {0}^{+} and \text{ln}(x)\to \infty as x\to \infty as shown in (Figure) and (Figure).

End behavior of the natural logarithm function
x 0.01 0.1 1 10 100
\text{ln}(x) -4.605 -2.303 0 2.303 4.605
The function f(x) = ln(x) is graphed.
Figure 22. The natural logarithm function approaches \infty as x\to \infty .

Determining End Behavior for a Transcendental Function

Find the limits as x\to \infty and x\to -\infty for f(x)=\frac{(2+3{e}^{x})}{(7-5{e}^{x})} and describe the end behavior of f.

Solution

To find the limit as x\to \infty , divide the numerator and denominator by {e}^{x}\text{:}

\begin{array}{cc}\hfill \underset{x\to \infty }{\text{lim}}f(x)& =\underset{x\to \infty }{\text{lim}}\frac{2+3{e}^{x}}{7-5{e}^{x}}\hfill \\ & =\underset{x\to \infty }{\text{lim}}\frac{(2\text{/}{e}^{x})+3}{(7\text{/}{e}^{x})-5}.\hfill \end{array}

As shown in (Figure), {e}^{x}\to \infty as x\to \infty . Therefore,

\underset{x\to \infty }{\text{lim}}\frac{2}{{e}^{x}}=0=\underset{x\to \infty }{\text{lim}}\frac{7}{{e}^{x}}.

We conclude that \underset{x\to \infty }{\text{lim}}f(x)=-\frac{3}{5}, and the graph of f approaches the horizontal asymptote y=-\frac{3}{5} as x\to \infty . To find the limit as x\to -\infty , use the fact that {e}^{x}\to 0 as x\to -\infty to conclude that \underset{x\to \infty }{\text{lim}}f(x)=\frac{2}{7}, and therefore the graph of approaches the horizontal asymptote y=\frac{2}{7} as x\to -\infty .

Find the limits as x\to \infty and x\to -\infty for f(x)=\frac{(3{e}^{x}-4)}{(5{e}^{x}+2)}.

Hint

\underset{x\to \infty }{\text{lim}}{e}^{x}=\infty and \underset{x\to -\infty }{\text{lim}}{e}^{x}=0.

Solution

\underset{x\to \infty }{\text{lim}}f(x)=\frac{3}{5}, \underset{x\to -\infty }{\text{lim}}f(x)=-2

Key Concepts

  • The limit of f(x) is L as x\to \infty (or as x\to -\infty ) if the values f(x) become arbitrarily close to L as x becomes sufficiently large.
  • The limit of f(x) is \infty as x\to \infty if f(x) becomes arbitrarily large as x becomes sufficiently large. The limit of f(x) is -\infty as x\to \infty if f(x)<0 and |f(x)| becomes arbitrarily large as x becomes sufficiently large. We can define the limit of f(x) as x approaches -\infty similarly.

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

1. The function graphed decreases very rapidly as it approaches x = 1 from the left, and on the other side of x = 1, it seems to start near infinity and then decrease rapidly.

Solution

x=1

2. The function graphed increases very rapidly as it approaches x = −3 from the left, and on the other side of x = −3, it seems to start near negative infinity and then increase rapidly to form a sort of U shape that is pointing down, with the other side of the U being at x = 2. On the other side of x = 2, the graph seems to start near infinity and then decrease rapidly.
3. The function graphed decreases very rapidly as it approaches x = −1 from the left, and on the other side of x = −1, it seems to start near negative infinity and then increase rapidly to form a sort of U shape that is pointing down, with the other side of the U being at x = 2. On the other side of x = 2, the graph seems to start near infinity and then decrease rapidly.

Solution

x=-1,x=2

4. The function graphed decreases very rapidly as it approaches x = 0 from the left, and on the other side of x = 0, it seems to start near infinity and then decrease rapidly to form a sort of U shape that is pointing up, with the other side of the U being at x = 1. On the other side of x = 1, there is another U shape pointing down, with its other side being at x = 2. On the other side of x = 2, the graph seems to start near negative infinity and then increase rapidly.
5. The function graphed decreases very rapidly as it approaches x = 0 from the left, and on the other side of x = 0, it seems to start near infinity and then decrease rapidly to form a sort of U shape that is pointing up, with the other side being a normal function that appears as if it will take the entirety of the values of the x-axis.

Solution

x=0

For the following functions f(x), determine whether there is an asymptote at x=a. Justify your answer without graphing on a calculator.

6. f(x)=\frac{x+1}{{x}^{2}+5x+4},a=-1

7. f(x)=\frac{x}{x-2},a=2

Solution

Yes, there is a vertical asymptote

8. f(x)={(x+2)}^{3\text{/}2},a=-2

9. f(x)={(x-1)}^{-1\text{/}3},a=1

Solution

Yes, there is vertical asymptote

10. f(x)=1+{x}^{-2\text{/}5},a=1

For the following exercises, evaluate the limit.

11. \underset{x\to \infty }{\text{lim}}\frac{1}{3x+6}

Solution

0

12. \underset{x\to \infty }{\text{lim}}\frac{2x-5}{4x}

13. \underset{x\to \infty }{\text{lim}}\frac{{x}^{2}-2x+5}{x+2}

Solution

\infty

14. \underset{x\to -\infty }{\text{lim}}\frac{3{x}^{3}-2x}{{x}^{2}+2x+8}

15. \underset{x\to -\infty }{\text{lim}}\frac{{x}^{4}-4{x}^{3}+1}{2-2{x}^{2}-7{x}^{4}}

Solution

-\frac{1}{7}

16. \underset{x\to \infty }{\text{lim}}\frac{3x}{\sqrt{{x}^{2}+1}}

17. \underset{x\to -\infty }{\text{lim}}\frac{\sqrt{4{x}^{2}-1}}{x+2}

Solution

-2

18. \underset{x\to \infty }{\text{lim}}\frac{4x}{\sqrt{{x}^{2}-1}}

19. \underset{x\to -\infty }{\text{lim}}\frac{4x}{\sqrt{{x}^{2}-1}}

Solution

-4

20. \underset{x\to \infty }{\text{lim}}\frac{2\sqrt{x}}{x-\sqrt{x}+1}

21. \underset{x\to \infty }{\text{lim}}\frac{3-x}{\sqrt{x-2}}

Solution

-\infty

22. \underset{x\to \infty }{\text{lim}}(\sqrt{x^2+5x-1}-x)

23. \underset{x\to -\infty }{\text{lim}}(\sqrt{x^2 +4x}+x)

Solution

-2

24. \underset{x\to \infty }{\text{lim}}(\sqrt{4x^2+3x}-2x)

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

25. f(x)=x-\frac{9}{x}

Solution

Horizontal: none, vertical: x=0

26. f(x)=\frac{1}{1-{x}^{2}}

27. f(x)=\frac{{x}^{3}}{4-{x}^{2}}

Solution

Horizontal: none, vertical: x=\pm 2

28. f(x)=\frac{{x}^{2}+3}{{x}^{2}+1}

29. f(x)= \sin (x) \sin (2x)

Solution

Horizontal: none, vertical: none

30. f(x)= \cos x+ \cos (3x)+ \cos (5x)

31. f(x)=\frac{x \sin (x)}{{x}^{2}-1}

Solution

Horizontal: y=0, vertical: x=\pm 1

32. f(x)=\frac{x}{ \sin (x)}

33. f(x)=\frac{1}{{x}^{3}+{x}^{2}}

Solution

Horizontal: y=0, vertical: x=0 and x=-1

34. f(x)=\frac{1}{x-1}-2x

35. f(x)=\frac{{x}^{3}+1}{{x}^{3}-1}

Solution

Horizontal: y=1, vertical: x=1

36. f(x)=\frac{ \sin x+ \cos x}{ \sin x- \cos x}

37. f(x)=x- \sin x

Solution

Horizontal: none, vertical: none

38. f(x)=\frac{1}{x}-\sqrt{x}

39. f(x) = \frac{x}{\sqrt[3]{x^3+1}}

Solution

Horizontal: y=1 , vertical: x=-1

40. f(x)= \frac{x-2}{\sqrt{3x^2+2x-1}}

For the following exercises, construct a function f(x) that has the given asymptotes.

41. x=1 and y=2

Solution

Answers will vary, for example: y=\frac{2x}{x-1}

42. x=1 and y=0

43. y=4 and x=-1

Solution

Answers will vary, for example: y=\frac{4x}{x+1}

44. x=0

For the following exercises, graph the function on a graphing calculator on the window x=\left[-5,5\right] and estimate the horizontal asymptote or limit. Then, calculate the actual horizontal asymptote or limit.

45. [T]f(x)=\frac{1}{x+10}

Solution

y=0

46. [T]f(x)=\frac{x+1}{{x}^{2}+7x+6}

47. [T]\underset{x\to -\infty }{\text{lim}}{x}^{2}+10x+25

Solution

\infty

48. [T]\underset{x\to -\infty }{\text{lim}}\frac{x+2}{{x}^{2}+7x+6}

49. [T]\underset{x\to \infty }{\text{lim}}\frac{3x+2}{x+5}

Solution

y=3

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

Glossary

end behavior
the behavior of a function as x\to \infty and x\to -\infty
horizontal asymptote
if \underset{x\to \infty }{\text{lim}}f(x)=L or \underset{x\to -\infty }{\text{lim}}f(x)=L, then y=L is a horizontal asymptote of f
infinite limit at infinity
a function that becomes arbitrarily large as x becomes large
limit at infinity
the limiting value, if it exists, of a function as x\to \infty or x\to -\infty
oblique asymptote
the line y=mx+b if f(x) approaches it as x\to \infty or x\to -\infty

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