2.7 Limits at Infinity

So far we have considered limits as \(x\) approaches a number \(c\). It is also important to consider limits where \(x\) approaches \(\infty\) or \(-\infty\), which we refer to as limits at infinity. In applications, limits at infinity arise naturally when we describe the “long-term” behavior of a system.

The notation \(x\rightarrow\infty\) indicates that \(x\) increases without bound, and \(x\rightarrow -\infty\) indicates that \(x\) decreases (through negative values) without bound. We write

• \(\lim\limits_{x\rightarrow \infty} f(x) = L\) if \(f(x)\) gets closer and closer to \(L\) as \(x\rightarrow\infty\).
• \(\lim\limits_{x\rightarrow -\infty} f(x) = L\) if \(f(x)\) gets closer and closer to \(L\) as \(x\rightarrow -\infty\).

As before, “closer and closer” means that \(|f(x)-L|\) becomes arbitrarily small. In either case, the line \(y=L\) is called a horizontal asymptote. We use the notation \(x\rightarrow\pm\infty\) to indicate that we are considering both infinite limits, as \(x\rightarrow\infty\) and as \(x\rightarrow -\infty\).

Infinite limits describe the asymptotic behavior of a function, which is behavior of the graph as we move out to the right or the left.

Example 1

Discuss the asymptotic behavior in Figure 2.41.

Figure 2.41: The lines \(y=7\) and \(y = 3\) are horizontal asymptotes of \(g(x)\).

Solution The function \(g(x)\) approaches \(L=7\) as we move to the right and it approaches \(L=3\) as we move to left, so

\[\lim\limits_{x\rightarrow \infty} g(x) = 7 \text{ and } \lim\limits_{x\rightarrow -\infty} g(x) = 3\]

Accordingly, the lines \(y=7\) and \(y = 3\) are horizontal asymptotes of \(g(x)\).

A function may approach an infinite limit as \(x\rightarrow\pm\infty\). We write

if \(f(x)\) becomes arbitrarily large as \(x\rightarrow\pm\infty\) respectively. Similar notation is used if \(f(x)\) approaches \(-\infty\) as \(x\rightarrow\pm\infty\). For example, we see in Figure 2.42(A) that

Figure 2.42: The above graphs display different possibilities of behaviors of functions as \(x\rightarrow\pm\infty\).

However, limits at infinity do not always exist. For example, \(f(x)=\sin x\) oscillates indefinitely [Figure 2.42(B)], so

do not exist.

The limits at infinity of the power functions \(f(x)=x^n\) are easily determined. If \(n > 0\), then \(x^n\) certainly increases without bound as \(x\rightarrow\infty\), so (Figure 2.43)

Figure 2.43: The given graphs depict some of the possible long-term bahaviours of power functions.

To describe the limits as \(x\rightarrow-\infty\), assume that \(n\) is a whole number so that \(x^n\) is defined for \(x < 0\). If \(n\) is even, then \(x^n\) becomes large and positive as \(x\rightarrow-\infty\), and if \(n\) is odd, it becomes large and negative. In summary,

The Basic Limit Laws (Theorem 1 in Section 2.7) are valid for limits at infinity. For example, the Sum and Constant Multiple Laws yield:

In general, if

where \(a_n,b_m\neq 0\), divide the numerator and denominator by \(x^m\):

The quotient in parenthesis approaches the finite limit \(\frac{a_n}{b_m}\) because

Therefore,

Here are some examples: \[\begin{aligned} \bullet & n=m: & \lim\limits_{x\rightarrow\infty}\frac{3x^4 - 7x + 9}{7x^4 - 4} &= \frac{3}{7}\lim\limits_{x\rightarrow\infty} x^0 = \frac{3}{7} \\ \bullet & n < m: & \lim\limits_{x\rightarrow\infty}\frac{3x^3 - 7x + 9}{7x^4 - 4} &= \frac{3}{7}\lim\limits_{x\rightarrow\infty} x^{-1} = 0 \\ \bullet & n > m, n-m \text{ odd}: & \lim\limits_{x\rightarrow-\infty}\frac{3x^8 - 7x + 9}{7x^3 - 4} &= \frac{3}{7}\lim\limits_{x\rightarrow-\infty} x^5 = -\infty \\ \bullet & n > m, n-m \text{ even}: & \lim\limits_{x\rightarrow-\infty}\frac{3x^7 - 7x + 9}{7x^3 - 4} &= \frac{3}{7}\lim\limits_{x\rightarrow\infty} x^4 = \infty \\ \end{aligned}\]

Our method can be adapted to noninteger exponents and algebraic functions.

Example 4

Calculate the limits at infinity of \(f(x)=\dfrac{12x+25}{\sqrt{16x^2+100x+500}}\).

Solution Divide numerator and denominator by \(x\) (multiply by \(x^{-1}\)), but notice the difference between \(x\) positive and \(x\) negative. For \(x > 0\),

\[\begin{aligned}x^{-1}\sqrt{16x^2+100x+500} & = \sqrt{x^{-2}}\sqrt{16x^2+100x+500} = \sqrt{16+{100\over x}+{500\over x^2}}\\ \lim\limits_{x\rightarrow \infty} \dfrac{12x+25}{\sqrt{16x^2+100x+500}} & = \dfrac{\lim\limits_{x\rightarrow \infty}12+{25\over x}}{\lim\limits_{x\rightarrow \infty}\sqrt{16+{100\over x}+{500\over x^2}}} = {12\over\sqrt{16}} = 3\end{aligned}\]

104

However, if \(x < 0\), then \(x=-\sqrt{x^2}\) and

\[x^{-1}\sqrt{16x^2+100x+500} = -\sqrt{x^{-2}}\sqrt{16x^2+100x+500} = -\sqrt{16+{100\over x}+{500\over x^2}}\]

So the limit as \(x\rightarrow-\infty\) is \(-3\) instead of \(3\) (Figure 2.44):

\[\lim\limits_{x\rightarrow \infty} \dfrac{12x+25}{\sqrt{16x^2+100x+500}} = \dfrac{\lim\limits_{x\rightarrow \infty}12+{25\over x}}{-\lim\limits_{x\rightarrow \infty}\sqrt{16+{100\over x}+{500\over x^2}}} = {12\over-\sqrt{16}} = -3\]

Figure 2.44: Graph of \(f(x) = \frac{12x+25}{\sqrt{16x^2+100x+500}}\) and its horizontal asymptotes \(y=-3\) and \(y=3\).

2.7.1 Section 2.7 Summary

• Limits as infinity:

• A horizontal line \(y=L\) is a horizontal asymptote if \[\lim\limits_{x\rightarrow \infty}f(x) = L\text{ and/or }\lim\limits_{x\rightarrow-\infty}f(x) = L\]
• If \(n > 0\), then \(\lim\limits_{x\rightarrow \infty}x^n = \infty\) and \(\lim\limits_{x\rightarrow\pm\infty}x^{-n} = 0\). If \(n > 0\) is a whole number, then \[\lim\limits_{x\rightarrow-\infty} x^n = \left\{\begin{aligned}\infty & \text{if }n\text{ is even}\\-\infty & \text{if }n\text{ is odd}\\ \end{aligned}\right.\text{ and } \lim\limits_{x\rightarrow \pm\infty} x^{-n} = 0\]
• If \(f(x)=\dfrac{a_nx^n + a_{n-1}x^{n-1}+\cdots+a_0}{b_mx^m + b_{m-1}x^{m-1}+\cdots+b_0}\) with \(a_n,b_m\neq 0\), then \[\lim\limits_{x\rightarrow\pm\infty}f(x) = \frac{a_n}{b_m}\lim\limits_{x\rightarrow\pm\infty}x^{n-m}\]