# How do you find all the asymptotes for function y = (x^2-x-6) /( x-2)?

Oct 23, 2015

$y = \frac{{x}^{2} - x - 6}{x - 2}$

has a vertical asymptote $x = 2$

and an oblique asymptote $y = x + 1$

#### Explanation:

$y = \frac{{x}^{2} - x - 6}{x - 2}$

$= \frac{{x}^{2} - 2 x + x - 2 - 4}{x - 2}$

$= \frac{x \left(x - 2\right) + \left(x - 2\right) - 4}{x - 2}$

$= x + 1 - \frac{4}{x - 2}$

As $x \to \pm \infty$, $\frac{4}{x - 2} \to 0$

So $y = x + 1$ is an oblique asymptote.

When $x = 2$, the denominator $\left(x - 2\right)$ is zero but the numerator $\left({x}^{2} - x - 6\right)$ is non-zero.

So $x = 2$ is a vertical asymptote.