Question

How do you find vertical, horizontal and oblique asymptotes for `(x^2)/(x-1)`?

Answer 1

Vertical asymptote at `x=1`

Oblique asymptote at `y = x+1`

Explanation:

A vertical asymptote occurs where the denominator is equal to `0`.

So we have:

`x-1=0->x=1`

So a vertical asymptote will occur where `x=1`.

The degree of the numerator is greater than the degree of the denominator so the function will not have horizontal asymptotes but will have oblique ones. To find them: we must split the fraction up like so:

`x^2/(x-1)=x+x/(x-1)=x+1+1/(x-1)`

So for very large values of `x` the fraction term will become insignificantly small and the function will approach the oblique asymptote: `y = x+1`.