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

Mar 27, 2016

vertical asymptote x = -4
horizontal asymptote y = 3

#### Explanation:

Vertical asymptotes occur as the denominator of a rational function tends to zero. To find the equation , let the denominator equal zero.

solve : x + 4 = 0 → x = -4 is the asymptote

Horizontal asymptotes occur as lim_(xto+-oo) f(x) → 0

divide terms on numerator/denominator by x

$\frac{\frac{3 x}{x}}{\frac{x}{x} + \frac{4}{x}} = \frac{3}{1 + \frac{4}{x}}$

as $x \to \infty , \frac{4}{x} \to 0$

$\Rightarrow y = \frac{3}{1} = 3 \text{ is the asymptote }$

Here is the graph of the function.
graph{(3x)/(x+4) [-20, 20, -10, 10]}