# How do you find the vertical, horizontal or slant asymptotes for  f(x) = (3x - 12 ) / ( x + 4) ?

Jun 4, 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 set the denominator equal to zero.

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

Horizontal asymptotes occur as

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

divide terms on numerator/denominator by x

$\frac{\frac{3 x}{x} - \frac{12}{x}}{\frac{x}{x} + \frac{4}{x}} = \frac{3 - \frac{12}{x}}{1 + \frac{4}{x}}$

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

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

Slant asymptotes occur when the degree of the numerator > degree of the denominator. This is not the case here (both of degree 1). Hence there are no slant asymptotes.
graph{(3x-12)/(x+4) [-20, 20, -10, 10]}