# For what values of x, if any, does f(x) = 1/((x-3)(x^2-27))  have vertical asymptotes?

Apr 12, 2016

$x = 3$,
$x = 3 \sqrt{3} = 5.2$ and
$x = - 3 \sqrt{3} = - 5.2$

#### Explanation:

Vertical asymptotes of a ratio of functions are identified by zeros of the denominator.

In the given function, denominator is $\left(x - 3\right) \left({x}^{2} - 27\right)$ or $\left(x - 3\right) \left(x - 3 \sqrt{3}\right) \left(x + 3 \sqrt{3}\right)$and hence three asymptotes are

$x = 3$, $x = 3 \sqrt{3} = 5.2$ and $x = - 3 \sqrt{3} = - 5.2$

graph{1/(x^3-3x^2-27x+81) [-10, 10, -2, 2]}