Question

How do you find the asymptotes for `Y=(3x)/(x^2-x-6) + 3`?

Answer 1

Horizontal asymptote is `y=3`
I have taken you to a point where you should be able to take over for the vertical ones.

Explanation:

As `abs(x)` increases then `abs(x^2)`becomes significantly larger and the other values of the fraction become more an more insignificant

`color(blue)("Horizontal asymptote")`

`lim_(abs(x)->oo) [(3x)/(x^2-x-6) + 3] =lim_(abs(x)->oo) [(3x)/(x^2-x-6)] +3 =3`

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
The equation becomes undefined as the fractions denominator approaches 0. So it is a matter of solving
`" "x^2-x-6=0`

`" "(x+2)(x-3)`

`color(blue)("Vertical asymptotes at "x=-2" and "x=+3)`