Chapter 12.4, Problem 59E

To determine

Whether the given statement “Every rational function has a horizontal asymptotes” is true or false.

Answer to Problem 59E

The given statement is false.

Explanation of Solution

Given information :

Given statement “Every rational function has a horizontal asymptote”

The given statement is false as every rational function need not have a horizontal asymptotes.

For example, the function $f(x)=\left(\frac{x^2}{x+3}\right)$ has no horizontal asymptotes as $\begin{array}{l}\underset{x\to \infty }{\lim }f(x)=\underset{x\to \infty }{\lim }\left(\frac{x^2}{x+3}\right)\\ =\underset{x\to \infty }{\lim }\left(\frac{x^2}{x\left(1+\frac{3}{x}\right)}\right)\\ =\underset{x\to \infty }{\lim }\frac{x}{\left(1+0\right)}\left(\because \underset{x\to \infty }{\lim }\frac{1}{x}=0\right)\\ \therefore \underset{x\to \infty }{\lim }f(x)=\infty \end{array}$ .

Which implies the limit does not exist as $x\to \infty$ .

Thus, there exist no horizontal asymptotes for the function $f(x)=\left(\frac{x^2}{x+3}\right)$ .

Hence, the given statement is false.