Chapter 2, Problem 165RE

To determine

To discuss whether every rational function has a vertical asymptote or not

Answer to Problem 165RE

Every rational function does not necessary have a vertical asymptote

Explanation of Solution

Given information:

A rational function

Consider a rational function $f(x)=\frac{P(x)}{Q(x)}$ , where $P$ and $Q$ are polynomial functions of $x$

and $Q(x)\ne 0$

  $f(x)$ has an asymptote which can be horizontal, vertical, or slanted that approaches the function but never touches.

The equation of vertical asymptote can be determined using equation $Q(x)=0$

Now, consider a function $g(x)=\frac{x+2}{x^2+9}$

The vertical asymptote of $g(x)$ can be determined by using equation $x^2+9=0$

  $x^2=-9$

This equation has no real solution,

Hence, $g(x)$ does not have a vertical asymptote.

  $\therefore$ Every rational function does not necessary have a vertical asymptote.