Question

How do you find the Vertical, Horizontal, and Oblique Asymptote given `2 /(x^2 + x -1)`?

Answer 1

The vertical asymptotes are `x=(-1+sqrt5)/2` and `x=(-1-sqrt5)/2`
The horizontal asymptote is `y=0`

Explanation:

Let's find the roots of the denominator
`x^2+x-1=0`
`Delta=1-(4*1*-1)=5`
So the roots are `(-1+-sqrt5)/2`
As we cannot divide by `0`, so the vertical asymptotes are
`x=(-1+sqrt5)/2` and `x=(-1-sqrt5)/2`
The limit of the expression as `x->+-oo` is `=0`
So the horizontal asymptote is `y=0`
And as the degree of the numerator is less than the degree of the denominator, ther is no oblique asymptote
graph{2/(x^2+x-1) [-11.25, 11.25, -5.62, 5.62]}