Question

What are the asymptotes of `f(x)=-x/((x-1)(3-x))`?

Answer 1

`f(x)` has the following asymptotes:
Vertical at `x=1` and `x=3`
Horizontal at `y=0` as `x->oo` and at `y=0` as `x->-oo`

Explanation:

`f(x)=-x/((x-1)(3-x))`

We will have vertical asymptotes when the denominator is zero,
ie
`(x-1)(3-x) = 0 => x=1,3`

As `x->oo => f(x) ~-x/(x(-x))`,
ie `f(x) ~1/x->0^+`

Similarly, As `x->-oo => f(x) -> 0^-`,

`f(x)` has the following asymptotes:
Vertical at `x=1` and `x=3`
Horizontal at `y=0` as `x->oo` and at `y=0` as `x->-oo`

graph{-x/((x-1)(3-x)) [-10, 10, -5, 5]}