# What are the asymptotes of f(x)=-x/((x^2-8)(9x-2) ?

Nov 22, 2016

Horizontal asymptote: at $y = 0$ because the degree on top is lower than the degree on bottom.

Vertical asymptotes:
$x = 2 \sqrt{2}$
$x = - 2 \sqrt{2}$
$x = \frac{2}{9}$

#### Explanation:

$f \left(x\right) = \frac{- x}{\left({x}^{2} - 8\right) \left(9 x - 2\right)}$

Factor completely:
$f \left(x\right) = \frac{- x}{\left(x - \sqrt{8}\right) \left(x + \sqrt{8}\right) \left(9\right) \left(x - \frac{2}{9}\right)}$

$f \left(x\right) = \frac{- x}{9 \left(x - 2 \sqrt{2}\right) \left(x + 2 \sqrt{2}\right) \left(x - \frac{2}{9}\right)}$

Vertical asymptotes occur when the denominator is equal to zero.
Vertical Asymptotes:
$x = 2 \sqrt{2}$
$x = - 2 \sqrt{2}$
$x = \frac{2}{9}$