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

1 Answer
Jan 1, 2018

Answer:

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#=># 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]#