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

Nov 8, 2016

vertical asymptote at x = 1
oblique asymptote at y = x + 1

#### Explanation:

The denominator of y cannot be zero as this would make y undefined. Equating the denominator to zero and solving gives the value that x cannot be and if the numerator is non-zero for this value then it is a vertical asymptote.

solve: $x - 1 = 0 \Rightarrow x = 1 \text{ is the asymptote}$

Horizontal asymptotes occur when the degree of the numerator ≤ degree of the denominator. This is not the case here ( numerator-degree 2 , denominator-degree 1 ) Hence there are no horizontal asymptotes.

Oblique asymptotes occur when the degree of the numerator > degree of the denominator. Hence there is an oblique asymptote.

Polynomial division gives.

$y = x + 1 + \frac{1}{x - 1}$

as $x \to \pm \infty , y \to x + 1 + 0$

$\Rightarrow y = x + 1 \text{ is the asymptote}$
graph{(x^2)/(x-1) [-10, 10, -5, 5]}