2.5 Limits at Infinity and Asymptotes

• •

  Calculate the limit of a function as $x$ increases or decreases without bound.

• •

  Recognize a horizontal asymptote on the graph of a function.

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 $f$ defined on an unbounded domain, we also need to know the behavior of $f$ as $x\to \pm \mathrm{\infty }$. 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 $f$.

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 Section 2.3, we looked at vertical asymptotes; in this section we deal with horizontal and oblique asymptotes.

Recall that $\underset{x\to a}{lim}f(x)=L$ 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 $f(x)=2+{\displaystyle \frac{1}{x}}$. As can be seen graphically in Figure 2.35 and numerically in Table 2.9, as the values of $x$ get larger, the values of $f(x)$ approach $2$. We say the limit as $x$ approaches $\mathrm{\infty }$ of $f(x)$ is $2$ and write $\underset{x\to \mathrm{\infty }}{lim}f(x)=2$. Similarly, for , as the values $|x|$ get larger, the values of $f(x)$ approaches $2$. We say the limit as $x$ approaches $-\mathrm{\infty }$ of $f(x)$ is $2$ and write $\underset{x\to -\mathrm{\infty }}{lim}f(x)=2$.

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

If the values $f(x)$ are getting arbitrarily close to some finite value $L$ as $x\to \mathrm{\infty }$ or $x\to -\mathrm{\infty }$, the graph of $f$ approaches the line $y=L$. In that case, the line $y=L$ is a horizontal asymptote of $f$ (Figure 2.36). For example, for the function $f(x)={\displaystyle \frac{1}{x}}$, since $\underset{x\to \mathrm{\infty }}{lim}f(x)=0$, the line $y=0$ is a horizontal asymptote of $f(x)={\displaystyle \frac{1}{x}}$.

A function cannot cross a vertical asymptote because the graph must approach infinity (or $-\mathrm{\infty })$ 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 $f(x)={\displaystyle \frac{(\cos x)}{x}}+1$ shown in Figure 2.37 intersects the horizontal asymptote $y=1$ 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 Section 2.3 also apply to limits at infinity. We illustrate how to use these laws to compute several limits at infinity.

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

For example, consider the function $f(x)=x^3$. As seen in Table 2.10 and Figure 2.42, as $x\to \mathrm{\infty }$ the values $f(x)$ become arbitrarily large. Therefore, $\underset{x\to \mathrm{\infty }}{lim}{x}^3=\mathrm{\infty }$. On the other hand, as $x\to -\mathrm{\infty }$, the values of $f(x)=x^3$ are negative but become arbitrarily large in magnitude. Consequently, $\underset{x\to -\mathrm{\infty }}{lim}{x}^3=-\mathrm{\infty }$.

The behavior of a function as $x\to \pm \mathrm{\infty }$ 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. 1.

   The function $f(x)$ approaches a horizontal asymptote $y=L$.

2. 2.

   The function $f(x)\to \mathrm{\infty }$ or $f(x)\to -\mathrm{\infty }$.

3. 3.

   The function does not approach a finite limit, nor does it approach $\mathrm{\infty }$ or $-\mathrm{\infty }$. 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.

Consider the power function $f(x)=x^n$ where $n$ is a positive integer. From Figure 2.43 and Figure 2.44, we see that

and

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

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

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

We now look at how the limits at infinity for power functions can be used to determine $\underset{x\to \pm \mathrm{\infty }}{lim}f(x)$ for any polynomial function $f$. Consider a polynomial function

of degree $n\ge 1$ so that $a_n\ne 0$. Factoring, we see that

As $x\to \pm \mathrm{\infty }$, all the terms inside the parentheses approach zero except the first term. We conclude that

For example, the function $f(x)=5x^3-3x^2+4$ behaves like $g(x)=5x^3$ as $x\to \pm \mathrm{\infty }$ as shown in Figure 2.45 and Table 2.11.

The end behavior for rational functions and functions involving radicals is a little more complicated than for polynomials. In Example 2.5.3, we show that the limits at infinity of a rational function $f(x)={\displaystyle \frac{p(x)}{q(x)}}$ 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 $x$ appearing in the denominator. This determines which term in the overall expression dominates the behavior of the function at large values of $x$.

Before proceeding, consider the graph of $f(x)={\displaystyle \frac{(3x^2+4x)}{(x+2)}}$ shown in Figure 2.49. As $x\to \mathrm{\infty }$ and $x\to -\mathrm{\infty }$, 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

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

Therefore, the graph of $f$ approaches the line $y=3x-2$ as $x\to \pm \mathrm{\infty }$. This line is known as an oblique asymptote for $f$ (Figure 2.49).

We can summarize the results of Example 2.5.3 to make the following conclusion regarding end behavior for rational functions. Consider a rational function

where $a_n\ne 0$ and $b_m\ne 0$.

1. 1.

   If the degree of the numerator is the same as the degree of the denominator $(n=m)$, then $f$ has a horizontal asymptote of $y=a_n/b_m$ as $x\to \pm \mathrm{\infty }$.

2. 2.

   If the degree of the numerator is less than the degree of the denominator , then $f$ has a horizontal asymptote of $y=0$ as $x\to \pm \mathrm{\infty }$.

3. 3.

   If the degree of the numerator is greater than the degree of the denominator $(n>m)$, then $f$ 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

   $$f(x)=\frac{p(x)}{q(x)}=g(x)+\frac{r(x)}{q(x)},$$

   where the degree of $r(x)$ is less than the degree of $q(x)$. As a result, $\underset{x\to \pm \mathrm{\infty }}{lim}r(x)/q(x)=0$. Therefore, the values of $[f(x)-g(x)]$ approach zero as $x\to \pm \mathrm{\infty }$. If the degree of $p(x)$ is exactly one more than the degree of $q(x)$$(n=m+1)$, the function $g(x)$ is a linear function. In this case, we call $g(x)$ an oblique asymptote.

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

The six basic trigonometric functions are periodic and do not approach a finite limit as $x\to \pm \mathrm{\infty }$. For example, $\sin x$ oscillates between $1$ and $-1$ (Figure 2.51). The tangent function $x$ has an infinite number of vertical asymptotes as $x\to \pm \mathrm{\infty };$ therefore, it does not approach a finite limit nor does it approach $\pm \mathrm{\infty }$ as $x\to \pm \mathrm{\infty }$ as shown in Figure 2.52.

Recall that for any base $b>0,b\ne 1$, the function $y=b^x$ is an exponential function with domain $(-\mathrm{\infty },\mathrm{\infty })$ and range $(0,\mathrm{\infty })$. If $b>1,y=b^x$ is increasing over $(-\mathrm{\infty },\mathrm{\infty })$. If , $y=b^x$ is decreasing over $(-\mathrm{\infty },\mathrm{\infty })$. For the natural exponential function $f(x)=e^x$, $e\approx 2.718>1$. Therefore, $f(x)=e^x$ is increasing on $(-\mathrm{\infty },\mathrm{\infty })$ and the range is $(0,\mathrm{\infty })$. The exponential function $f(x)=e^x$ approaches $\mathrm{\infty }$ as $x\to \mathrm{\infty }$ and approaches $0$ as $x\to -\mathrm{\infty }$ as shown in Table 2.12 and Figure 2.53.

Recall that the natural logarithm function $f(x)=\ln (x)$ is the inverse of the natural exponential function $y=e^x$. Therefore, the domain of $f(x)=\ln (x)$ is $(0,\mathrm{\infty })$ and the range is $(-\mathrm{\infty },\mathrm{\infty })$. The graph of $f(x)=\ln (x)$ is the reflection of the graph of $y=e^x$ about the line $y=x$. Therefore, $\ln (x)\to -\mathrm{\infty }$ as $x\to 0^{+}$ and $\ln (x)\to \mathrm{\infty }$ as $x\to \mathrm{\infty }$ as shown in Figure 2.54 and Table 2.13.

• •

  The limit of $f(x)$ is $L$ as $x\to \mathrm{\infty }$ (or as $x\to -\mathrm{\infty })$ if the values $f(x)$ become arbitrarily close to $L$ as $x$ becomes sufficiently large.

• •

  The limit of $f(x)$ is $\mathrm{\infty }$ as $x\to \mathrm{\infty }$ if $f(x)$ becomes arbitrarily large as $x$ becomes sufficiently large. The limit of $f(x)$ is $-\mathrm{\infty }$ as $x\to \mathrm{\infty }$ if and $|f(x)|$ becomes arbitrarily large as $x$ becomes sufficiently large. We can define the limit of $f(x)$ as $x$ approaches $-\mathrm{\infty }$ similarly.

• •

  For a polynomial function $p(x)=a_n{x}^n+a_{n-1}{x}^{n-1}+\mathrm{\dots }+a_{1}x+a_0$, where $a_n\ne 0$, the end behavior is determined by the leading term $a_n{x}^n$. If $n\ne 0$, $p(x)$ approaches $\mathrm{\infty }$ or $-\mathrm{\infty }$ at each end.

• •

  For a rational function $f(x)={\displaystyle \frac{p(x)}{q(x)}}$, the end behavior is determined by the relationship between the degree of $p$ and the degree of $q$. If the degree of $p$ is less than the degree of $q$, the line $y=0$ is a horizontal asymptote for $f$. If the degree of $p$ is equal to the degree of $q$, then the line $y={\displaystyle \frac{a_n}{b_n}}$ is a horizontal asymptote, where $a_n$ and $b_n$ are the leading coefficients of $p$ and $q$, respectively. If the degree of $p$ is greater than the degree of $q$, then $f$ approaches $\mathrm{\infty }$ or $-\mathrm{\infty }$ at each end.