# How do you find the asymptotes for (4x)/(x-3) ?

Apr 24, 2018

$\text{vertical asymptote at } x = 3$
$\text{horizontal asymptote at } y = 4$

#### Explanation:

$\text{let } f \left(x\right) = \frac{4 x}{x - 3}$

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.

$\text{solve "x-3=0rArrx=3" is the asymptote}$

$\text{Horizontal asymptotes occur as}$

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

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

$f \left(x\right) = \frac{\frac{4 x}{x}}{\frac{x}{x} - \frac{3}{x}} = \frac{4}{1 - \frac{3}{x}}$

$\text{as } x \to \pm \infty , f \left(x\right) \to \frac{4}{1 - 0}$

$\Rightarrow y = 4 \text{ is the asymptote}$
graph{(4x)/(x-3) [-20, 20, -10, 10]}