Limits at infinity and horizontal asymptotes

Section 3.6 Limits at infinity and horizontal asymptotes

We finally introduce one last type of limits, namely limits at infinity, and the related concept of horizontal asymptotes. These concern the behaviour of functions when you take the argument to be arbitrarily large (either positive or negative).

Objectives

You should be able to:

Evaluate limits at infinity.
Explain and illustrate the connection between limits at infinity and horizontal asymptotes.
Find the horizontal asymptotes, if any, of a function.

Subsection 3.6.1 Instructional video

Subsection 3.6.2 Key concepts

Concept 3.6.1. Limits at infinity.

Let \(f(x)\) be a function defined on some interval \((a, \infty)\text{.}\) Then

\begin{equation*} \lim_{x \to \infty} f(x) = L \end{equation*}

if the values of \(f(x)\) can be made arbitrarily close to \(L\) by requiring \(x\) to be sufficiently large.

Similarly, for a function \(f(x)\) defined on some interval \((-\infty, a)\text{,}\)

\begin{equation*} \lim_{x \to -\infty} f(x) = L \end{equation*}

if the values of \(f(x)\) can be made arbitrarily close to \(L\) by requiring \(x\) to be sufficiently large but negative.

In other words, we are looking at whether the function converges to a finite number \(L\) as \(x\) becomes very large either on the positive or negative side.

Note that this definition can be generalized to include limits at infinity that are infinite:

\begin{equation*} \displaystyle \lim_{x \to \infty} f(x) = \infty \end{equation*}

means that the values of \(f(x)\) can be made arbitrarily large for \(x\) sufficiently large. A similar definition holds for \(- \infty\text{.}\)

It can also happen that limits at infinity DNE, such as \(\displaystyle \lim_{x \to \infty} \sin (x)\text{.}\)

Concept 3.6.2. Some useful limits.

For all rational numbers \(r>0\text{,}\)

\begin{equation*} \lim_{x \to \infty} \frac{1}{x^r} = 0, \end{equation*}

and for all rational numbers \(r>0\) such that \(x^r\) is well defined for all \(x\text{,}\)

\begin{equation*} \lim_{x \to -\infty} \frac{1}{x^r} = 0. \end{equation*}

Concept 3.6.3. Tip to evaluate limits at infinity for rational functions.

To evaluate limits at infinity for rational functions (it also works for ratios of functions involving roots), divide both numerator and denominator by the largest power of \(x\) that occurs in the denominator.

Concept 3.6.4. Horizontal asymptotes.

The horizontal line \(y=L\) is a horizontal asymptote of \(y=f(x)\) if either

\begin{equation*} \lim_{x \to \infty} f(x) = L, \qquad \text{or} \qquad \lim_{x \to - \infty} f(x) = L. \end{equation*}

MATH 144: Calculus for the Physical Sciences I