# How do you find the vertical, horizontal or slant asymptotes for  f(x) = (3x) /( x+4)?

May 31, 2018

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

#### 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.

$\text{solve "x+4=0rArrx=-4" is the asymptote}$

$\text{horizontal asymptotes occur as}$

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

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

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

$\text{as } x \to \pm \infty , f \left(x\right) \to \frac{3}{1 + 0}$

$y = 3 \text{ is the asymptote}$

Slant asymptotes occur when the degree of the denominator is greater than the degree of the denominator. This is not the case here hence there are no slant asymptotes.
graph{(3x)/(x+4) [-20, 20, -10, 10]}