Question

How do you find the vertical, horizontal and slant asymptotes of: `Y=(3x)/(x^2-x-6) + 3`?

Answer 1

Horizontal asymptote y=3
Vertical asymptotess x=3 and x=-2
No slant asymptotes.

Explanation:

In case of rational functions, like the one given here, horizontal asymptotes are horizontal lines which the function approaches as `x->oo`. In the present case, as `x->oo` y=3. Hence y=3 is an horizontal asymptote of the given function.

For vertical asymptotes, in case of rational functions, it has to be seen that for what values of x, `y->oo`. In the present case if the denominator is factorised the function can be written as `y=x/((x-3)(x+2)) +3`. For x=3 and x=-2, `y->oo`. Hence vertical asymptotes are x=3 and x=-2.

There are no slant asymptotes.