# What are the asymptotes of y=x/(x^2-9) and how do you graph the function?

Mar 20, 2017

The vertical asymptotes are $x = - 3$ and $x = 3$
The horizontal asymptote is $y = 0$
No oblique asymptote

#### Explanation:

We need

${a}^{2} - {b}^{2} = \left(a + b\right) \left(a - b\right)$

We factorise the denominator

${x}^{2} - 9 = \left(x + 3\right) \left(x - 3\right)$

$y = \frac{x}{\left(x + 3\right) \left(x - 3\right)}$

As we cannot divide by $0$, x!=3 and $x \ne 3$

The vertical asymptotes are $x = - 3$ and $x = 3$

There are no oblique asymptotes as the degree of the numerator is $<$ than the degree of the denominator

${\lim}_{x \to - \infty} y = {\lim}_{x \to - \infty} \frac{x}{x} ^ 2 = {\lim}_{x \to - \infty} \frac{1}{x} = {0}^{-}$

${\lim}_{x \to + \infty} y = {\lim}_{x \to + \infty} \frac{x}{x} ^ 2 = {\lim}_{x \to + \infty} \frac{1}{x} = {0}^{+}$

The horizontal asymptote is $y = 0$

We can build a sign chart to have a general view of the graph

$\textcolor{w h i t e}{a a a a}$$x$$\textcolor{w h i t e}{a a a a}$$- \infty$$\textcolor{w h i t e}{a a a a}$$- 3$$\textcolor{w h i t e}{a a a a a a a a}$$0$$\textcolor{w h i t e}{a a a a a a a}$$+ 3$$\textcolor{w h i t e}{a a a a a a a}$$+ \infty$

$\textcolor{w h i t e}{a a a a}$$x + 3$$\textcolor{w h i t e}{a a a a}$$-$$\textcolor{w h i t e}{a a a}$$| |$$\textcolor{w h i t e}{a a a a}$$+$$\textcolor{w h i t e}{a a a a}$$+$$\textcolor{w h i t e}{a a a a a}$$| |$$\textcolor{w h i t e}{a a a}$$+$

$\textcolor{w h i t e}{a a a a}$$x$$\textcolor{w h i t e}{a a a a a a a a}$$-$$\textcolor{w h i t e}{a a a}$$| |$$\textcolor{w h i t e}{a a a a}$$-$$\textcolor{w h i t e}{a a a a}$$+$$\textcolor{w h i t e}{a a a a a}$$| |$$\textcolor{w h i t e}{a a a}$$+$

$\textcolor{w h i t e}{a a a a}$$x - 3$$\textcolor{w h i t e}{a a a a}$$-$$\textcolor{w h i t e}{a a a}$$| |$$\textcolor{w h i t e}{a a a a}$$-$$\textcolor{w h i t e}{a a a a}$$-$$\textcolor{w h i t e}{a a a a a}$$| |$$\textcolor{w h i t e}{a a a}$$+$

$\textcolor{w h i t e}{a a a a}$$y$$\textcolor{w h i t e}{a a a a a a a a}$$-$$\textcolor{w h i t e}{a a a}$$| |$$\textcolor{w h i t e}{a a a a}$$+$$\textcolor{w h i t e}{a a a a}$$-$$\textcolor{w h i t e}{a a a a a}$$| |$$\textcolor{w h i t e}{a a a}$$+$

The intercepts are $\left(0 , 0\right)$

${\lim}_{x \to - {3}^{-}} y = - \infty$

${\lim}_{x \to - {3}^{+}} y = + \infty$

${\lim}_{x \to {3}^{-}} y = - \infty$

${\lim}_{x \to {3}^{+}} y = + \infty$

Here is the graph

graph{(y-(x)/(x^2-9))(y)(y-1000(x+3))(y-1000(x-3))=0 [-18.05, 18.02, -9.01, 9.03]}