Chapter 12, Problem 108RE

To determine

To find: whether the statement is true or false.

Answer to Problem 108RE

The statement “limit of the rational function as $x$ approaches $\infty$ does not exist” is false

Explanation of Solution

Given information:

If the degree of the numerator $N\left(x\right)$ of a rational function $f\left(x\right)-N\left(x\right)/D\left(x\right)$ is greater than the degree of its denominator $D\left(x\right)$ , then the limit of the rational function as $x$ approaches $\infty$ is 0.

Calculation:

For the rational function $f\left(x\right)=\frac{N\left(x\right)}{D\left(x\right)}$ ,

Where $N\left(x\right)=a_n{x}^n+\mathrm{...}+a_0$ and $D\left(x\right)=b_m{x}^m+\mathrm{...}+b_0$ , the limits as $x$ approaches positive or negative infinity are as follows:

  $\underset{x\to \pm \infty }{\lim }f\left(x\right)=\{\begin{array}{l}0,\text{if}\text{n<m}\\ \frac{a_n}{b_m},\text{if}\text{n=m}\end{array}$

The limit does not exist, if $n>m$ .

The degree of the numerator is greater than the degree of the denominator. So, the limit of the function $f$ at infinity does not exist.

Consider the function $f\left(x\right)=\frac{a{x}^2+bx+c}{dx+e}$ .

Apply the limits,

  $\begin{array}{c}\underset{x\to 0}{\lim }f\left(x\right)=\underset{x\to 0}{\lim }\frac{a{x}^2+bx+c}{dx+e}\\ =\infty \end{array}$

Hence, the statement limit of the rational function as $x$ approaches $\infty$ does not exist is false.