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

Apr 16, 2016

vertical asymptote x = 0
oblique asymptote y = x

#### Explanation:

Vertical asymptotes occur as the denominator of a rational function tends to zero. To find the equation set the denominator equal to zero.

$\Rightarrow x = 0 \text{ is the asymptote }$

Horizontal asymptotes occur if the degree of the numerator ≤ degree of the denominator. This is not the case here hence there are no horizontal asymptotes.

Oblique asymptotes occur when the degree of the numerator > degree of the denominator. This is the case here hence there is an oblique asymptote.

To find equation divide numerator by denominator.

$\Rightarrow \frac{{x}^{2} - 1}{x} = {x}^{2} / x - \frac{1}{x} = x - \frac{1}{x}$

as $x \to \pm \infty , \frac{1}{x} \to 0 \text{ and } y \to x$

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