Question

What are the vertical and horizontal asymptotes for the following rational function: `r(x) = (x-2)/(x^2-8x-65)`?

Answer 1

vertical asymptotes x = -5 , x = 13
horizontal asymptote y = 0

Explanation:

The denominator of r(x) cannot be zero as this would be undefined. Equating the denominator to zero and solving gives the values that x cannot be and if the numerator is non-zero for these values then they are vertical asymptotes.

solve: `x^2-8x-65=0rArr(x-13)(x+5)=0`

`rArrx=-5,x=13" are the asymptotes"`

Horizontal asymptotes occur as

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

divide terms on numerator/denominator by the highest power of x, that is `x^2`

`(x/x^2-2/x^2)/(x^2/x^2-(8x)/x^2-65/x^2)=(1/x-2/x^2)/(1-8/x-65/x^2)`

as `xto+-oo,r(x)to(0-0)/(1-0-0)`

`rArry=0" is the asymptote"`
graph{(x-2)/(x^2-8x-65) [-20, 20, -10, 10]}