# How do you find the vertical, horizontal or slant asymptotes for y = x/(x-6)?

Apr 20, 2016

vertical asymptote x = 6
horizontal asymptote y = 1

#### Explanation:

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

solve: x - 6 = 0 → x = 6 , is the asymptote

Horizontal asymptotes occur as ${\lim}_{x \to \pm \infty} f \left(x\right) \to 0$

divide terms on numerator/denominator by x

$\Rightarrow \frac{\frac{x}{x}}{\frac{x}{x} - \frac{6}{x}} = \frac{1}{1 - \frac{6}{x}}$

as $x \to \pm \infty , \frac{6}{x} \to 0 \text{ and } y \to \frac{1}{1 - 0}$

$\Rightarrow y = 1 \text{ is the asymptote }$

Slant asymptotes occur when the degree of the numerator > degree of the denominator. This is not the case here hence there are no slant asymptotes.
graph{x/(x-6) [-20, 20, -10, 10]}