# How do you find the Vertical, Horizontal, and Oblique Asymptote given f(x)= 1/(x-2)?

Jul 31, 2016

vertical asymptote x = 2
horizontal asymptote y = 0

#### Explanation:

The denominator of f(x) cannot be zero as this is 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 - 2 = 0 → x = 2 is the asymptote

Horizontal asymptotes occur as

${\lim}_{x \to \pm \infty} , f \left(x\right) \to c \text{ (a constant)}$

divide terms on numerator/denominator by x

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

as $x \to \pm \infty , f \left(x\right) \to \frac{0}{1 - 0}$

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

Oblique asymptotes occur when the degree of the numerator > degree of the denominator. This is not the case here (numerator-degree 0 , denominator-degree 1 ) Hence there are no oblique asymptotes.
graph{(1)/(x-2) [-10, 10, -5, 5]}