Question

How do you find the horizontal asymptote for `(2x^2)/(x^2-4)`?

Answer 1

vertical asymptotes x = ± 2
horizontal asymptote y = 2

Explanation:

Vertical asymptotes occur as the denominator of a rational function tends to zero. To find the equation/s set the denominator equal to zero.

solve: `x^2-4=0rArr(x-2)(x+2)=0rArrx=±2`

`rArrx=-2,x=2" are the asymptotes"`

Horizontal asymptotes occur as

`lim_(xto+-oo),f(x)toc" (a constant)"`

divide terms on numerator/denominator by `x^2`

`((2x^2)/x^2)/(x^2/x^2-4/x^2)=2/(1-4/x^2)`

as `xto+-oo,f(x)to2/(1-0)`

`rArry=2" is the asymptote"`
graph{(2x^2)/(x^2-4) [-10, 10, -5, 5]}