Chapter 12.4, Problem 62E

To determine

Whether the given statement “when the degree of the numerator of a rational function is less than the degree of denominator, the limits at infinity do not exist” is true or false.

Answer to Problem 62E

The given statement false.

Explanation of Solution

Given information :

Given statement “when the degree of the numerator of a rational function is less than the degree of denominator, the limits at infinity do not exist”

The given statement is false as the degree of the numerator of a rational function is less than the degree of denominatorthen the limit always will exist and it equals to $0$ .

  $\begin{array}{l}For\text{example}f(x)=\frac{x+1}{x^2}\\ \Rightarrow \underset{x\to \infty }{\lim }f(x)=\underset{x\to \infty }{\lim }\frac{x\left(1+\frac{1}{x}\right)}{x^2}\\ =\underset{x\to \infty }{\lim }\frac{\left(1+0\right)}{x}\left(\because \underset{x\to \infty }{\lim }\frac{1}{x}=0\right)\\ \therefore \underset{x\to \infty }{\lim }f(x)=0\\ \Rightarrow \text{limit}\text{exist}\text{and}\text{equals}to0.\end{array}$

Hence, the given statement is false.