Question

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

Answer 1

the asymptotes are;
Vertical at `x=-sqrt(8),2,sqrt(8)`
Horizontal at `0` as `x->+-oo`

Explanation:

`f(x)=x/((x^2-8)(x-2))`
`:. f(x)=x/((x^2-sqrt(8)^2)(x-2))`
`:. f(x)=x/((x+sqrt(8))(x-sqrt(8))(x-2))`
So these will be vertical asymptotes.

So we can see that the denominator is zero when `x=-sqrt(8),2,sqrt(8)`

Also We have `f(x)=x/(x^3 + "lower terms")`
So as `x->+-oo => f(x)->0`

Hence, the asymptotes are;
Vertical at `x=-sqrt(8),2,sqrt(8)`
Horizontal at `0` as `x->+-oo`

The graph will help to visualise the results;
graph{x/((x^2-8)(x-2)) [-10, 10, -5, 5]}