# How do you identify all asymptotes for f(x)=(x-1)/(x-4)?

Sep 27, 2016

vertical asymptote at x = 4
horizontal asymptote at y = 1

#### Explanation:

The denominator of y cannot be zero as this would make y 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: $x - 4 = 0 \Rightarrow x = 4 \text{ is the asymptote}$

Horizontal asymptotes occur as

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

divide terms on numerator/denominator by x

$y = \frac{\frac{x}{x} - \frac{1}{x}}{\frac{x}{x} - \frac{4}{x}} = \frac{1 - \frac{1}{x}}{1 - \frac{4}{x}}$

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

$\Rightarrow y = 1 \text{ is the asymptote}$
graph{(x-1)/(x-4) [-10, 10, -5, 5]}