# How do you find all asymptotes and intercepts of f(x)=(5x+15)/(x^2-x-12)?

Jan 1, 2018

See explanation

#### Explanation:

f(x)=(5x+15)/(x^2−x−12)=(5(x+3))/((x-4)(x+3))=>**5/(x-4)**

D(f)=RR-{-3; 4}

Vertical asymptotes are usually in the points which aren't in the domain. x=-3 isn't asymptote because we can simplify f(x) to the form highlighted above (after substituting x for -3 we get finite value$\implies$ it isn't asymptote)

Horizontal asymptote: in $+ \infty$

y=Lim_(xrarroo)(5x+15)/(x^2−x−12)=Lim_(xrarroo)(5/x+15/x^2)/(1-1/x-12/x^2)=0/1=0
$y = 0$

Horizontal asymptote: in $- \infty$
it's the same: $y = 0$

Intercepts:
$\mathmr{if} x = 0 \quad \implies \quad y = - \frac{15}{12} = - \frac{5}{4}$

$\mathmr{if} y = 0 \quad \implies \quad 0 = 5 x + 15 \implies x = - \frac{15}{5} = - 3$

rarr[0;-5/4],quadquadquad[-3;0]