# How do you find all the asymptotes for function y=(x^2-4)/(x)?

The vertical asymptote is $x = 0$ and oblique asymptote
$y = x$

#### Explanation:

A line $x = a$ is a vertical asymptote of a function f(x) if

${\lim}_{x \to a} f \left(x\right) = \pm \infty$

A line $y = b$ is a horizontal asymptote of a function f(x) if

${\lim}_{x \to \pm \infty} = b$

An oblique asymptote for a function f(x) has the formula

$y = c x + d$

where

$c = {\lim}_{x \to \infty} f \frac{x}{x}$ and $d = {\lim}_{x \to \infty} \left(f \left(x\right) - c x\right)$

Hence we have that

$f \left(x\right) \to \pm \infty , x \to 0$

and $c = \lim f \frac{x}{x} = \lim \frac{{x}^{2} - 4}{x} ^ 2 = 1$

$d = \lim \left(\frac{{x}^{2} - 4}{x} - x\right) = 0$

Hence $x = 0$ and $y = x$