Question

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

Answer 1

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

Vertical asymptotes:
`x=2sqrt2`
`x=-2sqrt2`
`x=2/9`

Explanation:

`f(x)=frac{-x}{(x^2-8)(9x-2)}`

Factor completely:
`f(x)=frac{-x}{(x-sqrt8)(x+sqrt8)(9)(x-2/9)}`

`f(x)=frac{-x}{9(x-2sqrt2)(x+2sqrt2)(x-2/9)}`

Vertical asymptotes occur when the denominator is equal to zero.
Vertical Asymptotes:
`x=2sqrt2`
`x=-2sqrt2`
`x=2/9`