Question

How do you find the asymptotes for `f(x) = x / (3x(x-1))`?

Answer 1

The vertical asymptote is `x=1`
The horizontal asymptote is `y=0`

Explanation:

You can simplify `f(x)=1/(3(x-1))`
As you cannot divide by `0`, the vertical asymptote is `x=1`

There are no slant asymptotes since the degree of the numerator `<` the degree of the denominator

`lim_(x->-oo)f(x)=lim_(x->-oo)1/(3x)=0^(-)`

`lim_(x->+oo)f(x)=lim_(x->+oo)1/(3x)=0^(+)`

So `y=0` is a horizontal asymptote.
graph{1/(3(x-1)) [-7.9, 7.9, -3.95, 3.95]}