How do you find the Vertical, Horizontal, and Oblique Asymptote given f(x)= x/(x(x-2))?

May 1, 2016

vertical asymptote x = 2
horizontal asymptote y = 0

Explanation:

The first step here is to simplify f(x) by cancelling the x.

$\Rightarrow f \left(x\right) = \frac{\cancel{x}}{\cancel{x} \left(x - 2\right)} = \frac{1}{x - 2}$

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 - 2 = 0 → x = 2 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

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

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

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

Oblique asymptotes occur when the degree of the numerator > degree of the denominator. This is not the case here hence there are no oblique asymptotes.
graph{1/(x-2) [-10, 10, -5, 5]}