Chapter 11.4, Problem 64E

To determine

To evaluate: the whether the statement is true or false.

Answer to Problem 64E

The given statement is false.

Explanation of Solution

Given:

When the degrees of the numerator and denominator of a rational function are equal, then the limit as x approaches infinity does not exist.

The given statement is false.

For example, the function such as, $f\left(x\right)=\frac{x+1}{x+2}$ have equal degree in numerator and denominator.

Now, finding the limit,

  $\begin{array}{c}\underset{x\to \infty }{\lim }f\left(x\right)=\underset{x\to \infty }{\lim }\frac{x+1}{x+2}\\ =\underset{x\to \infty }{\lim }\frac{x\left(1+\frac{1}{x}\right)}{x\left(1+\frac{2}{x}\right)}\\ =1\end{array}$

Hence, the limit exists.

So, the given statement is false.