Question

How do you find vertical, horizontal and oblique asymptotes for `x / (3x(x-1))`?

Answer 1

There is a vertical asymptote at `x=1` and a horizontal asymptote at `y=0`

Explanation:

To find all the asymptotes for function `y=x/(3x(x-1))`, we first observe that `x` cancels out from numerator and denominator and the function is primarily `x/(3(x-1))`, but there is a hole at `x=0`

Let us first start with vertical asymptotes, which are given by putting denominator equal to zero or `x-1=0` i.e. `x=1`.

Further as in `y=3/(x-1)`, there is no variable in numerator, we have a horizontal asymptote at `y=0`

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