Question

How do you find the vertical, horizontal or slant asymptotes for `f(x) = (2x-8) / (3x-24)`?

Answer 1

vertical asymptote x = 8
horizontal asymptote `y=2/3`

Explanation:

The denominator of f(x) cannot be zero as this is undefined. Equating the denominator to zero and solving gives the value that x cannot be and if the numerator is non-zero for this value then it is a vertical asymptote.

solve: 3x - 24 = 0 → x = 8 is the asymptote

Horizontal asymptotes occur as

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

divide terms on numerator/denominator by x

`((2x)/x-8/x)/((3x)/x-24/x)=(2-8/x)/(3-24/x)`

as `xto+-oo,f(x)to(2-0)/(3-0)`

`rArry=2/3" is the asymptote"`

Slant asymptotes occur when the degree of the numerator > degree of the denominator. This is not the case her ( both degree 1 ) Hence there are no slant asymptotes.
graph{(2x-8)/(3x-24) [-20, 20, -10, 10]}