Question

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

Answer 1

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]`