Horizontal asymptotes

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We explore functions that behave like horizontal lines as the input grows without bound.

Consider the function: $f(x) = \frac {6x-9}{x-1}$

As $x$ approaches infinity, it seems like $f(x)$ approaches a specific value. Such a limit is called a limit at infinity.

If $f(x)$ becomes arbitrarily close to a specific value $L$ by making $x$ sufficiently large, we write $\lim _{x\to \infty } f(x) = L$ and we say, the limit at infinity of $f(x)$ is $L$ .

If $f(x)$ becomes arbitrarily close to a specific value $L$ by making $x$ sufficiently large and negative, we write $\lim _{x\to -\infty } f(x) = L$ and we say, the limit at negative infinity of $f(x)$ is $L$ .

Compute $\lim _{x\to \infty } \frac {6x-9}{x-1}.$

Write $\begin{align*} \lim_{x\to\infty}\frac{6x-9}{x-1} &= \lim_{x\to\infty}\frac{6x-9}{x-1}\cdot \frac{1/x}{1/x}\\ &=\lim_{x\to\infty}\frac{\frac{6x}{x} - \frac{9}{x}}{\frac{x}{x} - \frac{1}{x}}\\ &= \lim_{x\to\infty} \frac{6}{1}\\ &= 6. \end{align*}$

Sometimes one must be careful, consider this example.

Compute $\lim _{x\to -\infty } \frac {x^3+1}{\sqrt {x^6+5}}$

In this case we multiply the numerator and denominator by $-1/x^3$ , which is a positive number as since $x\to -\infty$ , $x^3$ is a negative number. $\begin{align*} \lim_{x\to -\infty} \frac{x^3+1}{\sqrt{x^6+5}} &= \lim_{x\to -\infty} \frac{x^3+1}{\sqrt{x^6+5}} \cdot \frac{-1/x^3}{-1/x^3}\\ &= \lim_{x\to -\infty} \frac{-1-1/x^3}{\sqrt{x^6/x^6+5/x^6}}\\ &= -1. \end{align*}$

Note, since $\lim _{x\to \infty } f(x) = \lim _{x\to 0^+} f\left (\frac {1}{x}\right )$ and $\lim _{x\to -\infty } f(x) = \lim _{x\to 0^-} f\left (\frac {1}{x}\right )$ we can also apply the Squeeze Theorem when taking limits at infinity. Here is an example of a limit at infinity that uses the Squeeze Theorem, and shows that functions can, in fact, cross their horizontal asymptotes.

If $\lim _{x\to \infty } f(x) = L \qquad \text {or}\qquad \lim _{x\to -\infty } f(x) = L,$ then the line $y=L$ is a horizontal asymptote of $f(x)$ .

Give the horizontal asymptotes of $f(x) = \frac {6x-9}{x-1}$

From our previous work, we see that $\lim _{x\to \infty } f(x) = 6$ , and upon further inspection, we see that $\lim _{x\to -\infty } f(x) = 6$ . Hence the horizontal asymptote of $f(x)$ is the line $y=6$ .

It is a common misconception that a function cannot cross an asymptote. As the next example shows, a function can cross a horizontal asymptote, and in the example this occurs an infinite number of times!

Give a horizontal asymptote of $f(x) = \frac {\sin (7x)+4x}{x}.$

Again from previous work, we see that $\lim _{x\to \infty } f(x) = \answer [given]{4}$ . Hence $y=\answer [given]{4}$ is a horizontal asymptote of $f(x)$ .

We conclude with an infinite limit at infinity.

Compute $\lim _{x\to \infty } \ln (x)$

The function $\ln (x)$ grows very slowly, and seems like it may have a horizontal asymptote, see the graph above. However, if we consider the definition of the natural log as the inverse of the exponential function

$\ln (x) = y$ means that $e^y =x$ and that $x$ is positive.

We see that we may raise $e$ to higher and higher values to obtain larger numbers. This means that $\ln (x)$ is unbounded, and hence $\lim _{x\to \infty }\ln (x)=\infty$ .