# How do you identify all asymptotes for f(x)=1/(x-1)?

Apr 2, 2017

vertical asymptote at x = 1
horizontal asymptote at y = 0

#### Explanation:

The denominator of f(x) cannot be zero as this would make f(x) 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.

$\text{solve "x-1=0rArrx=1" is the asymptote}$

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

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

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

$\Rightarrow y = 0 \text{ is the asymptote}$
graph{1/(x-1) [-10, 10, -5, 5]}