Question

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

Answer 1

`x=3`,
`x=3sqrt3=5.2` and
`x=-3sqrt3=-5.2`

Explanation:

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

In the given function, denominator is `(x-3)(x^2-27)` or `(x-3)(x-3sqrt3)(x+3sqrt3)`and hence three asymptotes are

`x=3`, `x=3sqrt3=5.2` and `x=-3sqrt3=-5.2`

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