OpenStax Calculus: Volume 1

Subsubsection 4.6.1.1 Limits at Infinity and Horizontal Asymptotes

Recall that \(\displaystyle{\lim_{x\to a}f(x)=L}\) 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 4.86 and numerically in Table 4.87, 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 \(\displaystyle{\lim_{x\to\infty}f(x)=2.}\) Similarly, for \(x\lt 0,\) 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 \(\displaystyle{\lim_{x\to a}f(x)=2.}\)

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 \(\displaystyle{\lim_{x\to\infty}f(x)=L.}\) 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 \(x\lt 0\) and \(|x|\) is sufficiently large. In that case, we write \(\displaystyle{\lim_{x\to-\infty}f(x)=L.}\) 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 \(x\to\infty\) 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 4.90). For example, for the function \(f(x)=\frac{1}{x},\) since \(\displaystyle{\lim_{x\to\infty}f(x)=0,}\) the line \(y=0\) is a horizontal asymptote of \(f(x)=\frac{1}{x}.\)

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. The function \(f(x)=\frac{-2x+5}{x^2-4x+3}\) intersects its horizontal asymptote \(y=0\) once as shown in Figure 4.91

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 4.92 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 we introduced in Introduction to Limits 2 also apply to limits at infinity. We illustrate how to use these laws to compute several limits at infinity. We will make use of the following theorem in order to evaluate limits at infinity.

Subsubsection 4.6.1.2 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 \(\displaystyle{\lim_{x\to\infty}f(x)=\infty}\) (or \(\displaystyle{\lim_{x\to-\infty}f(x)=\infty).}\) 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 \(\displaystyle{\lim_{x\to\infty}f(x)=-\infty}\) (or \(\displaystyle{\lim_{x\to-\infty}f(x)=-\infty).}\)

For example, consider the function \(f(x)=x^3.\) As seen in Table 4.97 and Figure 4.108, as \(x\to\infty\) the values \(f(x)\) become arbitrarily large. Therefore, \(\displaystyle{\lim_{x\to\infty}x^3=\infty.}\) On the other hand, as \(x\to-\infty,\) the values of \(f(x)=x^3\) are negative but become arbitrarily large in magnitude. Consequently, \(\displaystyle{\lim_{x\to-\infty}x^3=-\infty.}\)

Subsubsection 4.6.2.1 End Behavior for Polynomial Functions

Consider the power function \(f(x)=x^n\) where \(n\) is a positive integer. From Figure 4.100 and Figure 4.101, we see that

and

Using these facts, it is not difficult to evaluate \(\displaystyle{\lim_{x\to\infty}cx^n}\) and \(\displaystyle{\lim_{x\to-\infty}cx^n,}\) where \(c\) is any constant and \(n\) is a positive integer. If \(c\gt 0,\) the graph of \(y=cx^n\) is a vertical stretch or compression of \(y=x^n,\) and therefore

If \(c\lt 0,\) the graph of \(y=cx^n\) is a vertical stretch or compression combined with a reflection about the \(x\)-axis, and therefore

If \(c=0,y=cx^n=0,\) in which case \(\displaystyle{\lim_{x\to\infty}cx^n=0=\lim_{x\to-\infty}cx^n.}\)

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

of degree \(n\geq 1\) so that \(a_n\neq 0.\) Factoring, we see that

As \(x\to\pm\infty,\) 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 \(x\to\pm\infty\) as shown in Figure 4.104 and Table 4.105.

Subsubsection 4.6.2.2 End Behavior for Algebraic Functions

The end behavior for rational functions and functions involving radicals is a little more complicated than for polynomials. In Example 4.106, we show that the limits at infinity of a rational function \(f(x)=\frac{p(x)}{q(x)}\) 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 \(x.\)

We can summarize the results of Example 4.106 to make the following conclusion regarding end behavior for rational functions. Consider a rational function

where \(a_n\neq 0 \text{ and } b_m\neq 0.\)

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 \(x\to\pm\infty.\)

2. If the degree of the numerator is less than the degree of the denominator \((n\lt m),\) then \(f\) has a horizontal asymptote of \(y=0\) as \(x\to\pm\infty.\)

3. If the degree of the numerator is greater than the degree of the denominator \((n\gt m),\) 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

   \begin{equation*} f(x)=\frac{p(x)}{q(x)}=g(x)+\frac{r(x)}{q(x)}, \end{equation*}
   where the degree of \(r(x)\) is less than the degree of \(q(x).\) As a result, \(\displaystyle{\lim_{x\to\pm\infty}r(x)/q(x)=0.}\) Therefore, the values of \([f(x)-g(x)]\) approach zero as \(x\to\pm\infty.\) Now let's consider the end behavior for functions involving a radical.

Example 4.111. Determining End Behavior for a Function Involving a Radical.

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

\begin{equation*} \sqrt{4x^2+5}\approx \sqrt{4x^2}=2|x| \end{equation*}

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 \(x\gt 0,\) \(|x|=-x\) for \(x\lt 0,\) and \(|x|=\sqrt{x^2}\) for all \(x,\) we calculate the limits as follows:

\begin{align*} \lim_{x\to\infty}\frac{3x-2}{\sqrt{4x^2+5}}\amp = \lim_{x\to\infty}\frac{(1/|x|)}{(3x-2)}{(1/|x|)\sqrt{4x^2+5}}\\ \amp=\lim_{x\to\infty}\frac{(1/x)}{(3x-2)}{\sqrt{(1/x^2)}{(4x^2+5)}}\\ \amp= \lim_{x\to\infty}\frac{3-2/x}{\sqrt{4+5/x^2}}=\frac{3}{\sqrt{4}}=\frac{3}{2}\\ \lim_{x\to-\infty}\frac{3x-2}{\sqrt{4x^2+5}}\amp = \lim_{x\to-\infty}\frac{(1/|x|)}{(3x-2)}{(1/|x|)\sqrt{4x^2+5}}\\ \amp= \lim_{x\to-\infty}\frac{(-1/x)}{(3x-2)}{\sqrt{(1/x^2)}{(4x^2+5)}}\\ \amp= \lim_{x\to-\infty}\frac{-3+2/x}{\sqrt{4+5/x^2}}=\frac{-3}{\sqrt{4}}=\frac{-3}{2}. \end{align*}

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.

Subsubsection 4.6.2.3 Determining End Behavior for Transcendental Functions

Recall that for any base \(b\gt 0,b\neq 1,\) the function \(y=b^x\) is an exponential function with domain \((-\infty,\infty)\) and range \((0,\infty).\) If \(b\gt 1,y=b^x\) is increasing over \((-\infty,\infty).\) If \(0\lt b\lt 1,\) \(y=b^x\) is decreasing over \((-\infty,\infty).\) For the natural exponential function \(f(x)=e^x,\) \(e\approx 2.718\gt 1.\) 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 Table 4.114 and Figure 4.115.

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 \(x\to0^+\) and \(\text{ ln }(x)\to\infty\) as \(x\to\infty\) as shown in Figure 4.98 and Table 4.116.