# Chapter 2 - Section 2.6 - Rational Functions and Their Graphs - Exercise Set - Page 399: 25

Vertical asymptote is $x=-4$ and $x=0$ is a hole.

#### Work Step by Step

If there is a rational function $f\left( x \right)=\frac{p\left( x \right)}{q\left( x \right)}$ where $p\left( x \right)$ is the numerator and $q\left( x \right)$ is the denominator, then $x=a$ is a vertical asymptote of the function $f\left( x \right)$ if $x=a$ is a zero of the denominator $q\left( x \right)$. After cancelling the common factor $x$ from the numerator and denominator, the function $h\left( x \right)=\frac{1}{x+4}$ For the vertical asymptote, equate the denominator to zero. \begin{align} & x+4=0 \\ & x=-4 \end{align} Thus, $x=-4$ is a vertical asymptote. If there is a rational function $f\left( x \right)=\frac{p\left( x \right)}{q\left( x \right)}$, where $p\left( x \right)$ is the numerator and $q\left( x \right)$ is the denominator then $x=a$ is called a hole if $x-a$ is a common factor of the numerator and denominator. There is a common factor between $x$ and $x\left( x+4 \right)$ is $x$. So, $x=0$ is a hole. Hence, $x=-4$ is a vertical asymptote and $x=0\text{ is a hole}$.

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