Exercise 13 Page 526

We want find any holes, as well as the vertical and horizontal asymptotes, in the graph of the function.

Holes and Vertical Asymptotes

Let's start by factoring the denominator. Recall that division by zero is not defined. Therefore, the rational function is undefined where $(x-2)(x+5)=0.$ We will use the Zero Product Property to solve this equation. The function is not defined when $x=2,$ and $x=\text{-}5.$ This means we have either a hole or a vertical asymptote. We cannot simplify the function. This indicates the lines $x=2$ and $x=\text{-}5$ are vertical asymptotes.

Horizontal Asymptotes

Let's recall how to find horizontal asymptotes in rational functions.

$y=\frac{p(x)}{q(x)}$
The degree of $p(x)$ is greater than the degree of $q(x)$	There are no horizontal asymptotes
The degree of $q(x)$ is greater than the degree of $p(x)$	$y=0$ is a horizontal asymptote
The degrees of $p(x)$ and $q(x)$ are equal	$y=\frac{a}{b}$ is a horizontal asymptote, where $a$ and $b$ are the leading coefficients of $p(x)$ and $q(x),$ respectively

Let's now consider the given function. Since the degree of the denominator is greater than the degree of the numerator, the line $y=0$ is a horizontal asymptote.