Question

How do you find the Vertical, Horizontal, and Oblique Asymptote given `y = (2x^2 - 11)/( x^2 + 9)`?

Answer 1

Only a horizontal asymptote `y=2`.

Explanation:

As `y=(2x^2-11)/(x^2+9)=(2-11/x^2)/(1+9/x^2)`

`Lt_(x->+-oo)(2-11/x^2)/(1+9/x^2)=2`

Also observe that the denominator `x^2+9>0`, as `x^2` is always positive.

Hence we have only a horizontal asymptote `y=2`.
graph{(y-(2x^2-11)/(x^2+9))(y-2)=0 [-10, 10, -5, 5]}