How do you find all asymptotes and intercepts of f(x)=(5x+15)/(x^2-x-12)f(x)=5x+15x2x12?

1 Answer
Jan 1, 2018

See explanation

Explanation:

f(x)=(5x+15)/(x^2−x−12)=(5(x+3))/((x-4)(x+3))=>**5/(x-4)**f(x)=5x+15x2x12=5(x+3)(x4)(x+3)5x4

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=> it isn't asymptote)

Horizontal asymptote: in +oo

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 -oo
it's the same: y=0

Intercepts:
if x=0quad=>quady=-15/12=-5/4

if y=0quad=>quad0=5x+15=>x=-15/5=-3

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