Question

What are the asymptotes and removable discontinuities, if any, of `f(x)= (9x^2-36)/(x^2-9)`?

Answer 1

Vertical asymptotes at:`color(white)("XXX")x=3 and x=-3`
Horizontal asymptote at:`color(white)("XX")f(x)=9`
There are no removable discontinuities.

Explanation:

`f(x)=(x^2-36)/(x^2-9)`

`color(white)("XXX")=(9(x-2)(x+2))/((x-3)(x+3))`

Since the numerator and denominator have no common factors
there are no removable discontinuities

and the values which cause the denominator to become `0`
form vertical asymptotes:
`color(white)("XXX")x=3 and x=-3`

Noting
`color(white)("XXX")lim_(xrarroo)(x-2)/(x-3) = 1`
and
`color(white)("XXX")lim_(xrarroo)(x+2)/(x+3)=1`

`lim_(xrarroo) (9(x-2)(x+2))/((x-3)(x+3))=9`

So `f(x)=9` forms a horizontal asymptote.