# What are the asymptotes of f(x)=(1-5x)/(1+2x)?

Jul 29, 2017

$\text{vertical asymptote at } x = - \frac{1}{2}$
$\text{horizontal asymptote at } y = - \frac{5}{2}$

#### 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 verical asymptote.

$\text{solve "1+2x=0rArrx=-1/2" is the asymptote}$

$\text{horizontal asymptotes occur as}$

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

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

$f \left(x\right) = \frac{\frac{1}{x} - \frac{5 x}{x}}{\frac{1}{x} + \frac{2 x}{x}} = \frac{\frac{1}{x} - 5}{\frac{1}{x} + 2}$

as $x \to \pm \infty , f \left(x\right) \to \frac{0 - 5}{0 + 2}$

$\Rightarrow y = - \frac{5}{2} \text{ is the asymptote}$