Question

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

Answer 1

VA: `x=+-sqrt6`
HA: `x=2`

Explanation:

Vertical Asymptotes

These will occur when the denominator equals `0`.

`x^2-6=0`
`x^2=6`
`x=+-sqrt6`

Thus, there are vertical asymptotes at `x=-sqrt6` and `x=sqrt6`.

Horizontal Asymptotes

When the degree of the numerator and denominator are equal, like they are here, divide the terms with the greatest degree. In this case, those are `2x^2` and `x^2`.

`(2x^2)/x^2=2`

There is a horizontal asymptote at `x=2`.

graph{(2x^2+3)/(x^2-6) [-19.62, 20.93, -8.83, 11.45]}