# How do you find any asymptotes of f(x)=x/(x-5)?

May 16, 2018

VA: $x = 5$
HA: $y = 1$

#### Explanation:

(VA) Vertical Asymptote: Set the denominator equal to zero:

$x - 5 = 0$

$x = 5$

(HA) Horizontal Asymptote: Divide the coefficients of the x values:

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

$y = 1$

May 16, 2018

$\text{vertical asymptote at } x = 5$
$\text{horizontal asymptote at } y = 1$

#### 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.

$\text{solve "x-5=0rArrx=5" is the asymptote}$

$\text{horizontal asymptotes occur as}$

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

$\text{divide terms on numerator/denominator by x}$

$f \left(x\right) = \frac{\frac{x}{x}}{\frac{x}{x} - \frac{5}{x}} = \frac{1}{1 - \frac{5}{x}}$

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

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