Question

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

Answer 1

vertical asymptote x = 2
horizontal asymptote y = 0

Explanation:

The first step here is to simplify f(x) by cancelling the x.

`rArrf(x)= cancel(x)/(cancel(x) (x-2))=1/(x-2)`

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

solve : x - 2 = 0 → x = 2 is the asymptote

Horizontal asymptotes occur as `lim_(x to +- oo) f(x) to 0`

divide terms on numerator/denominator by x

`rArr (1/x)/(x/x-2/x)=(1/x)/(1-2/x)`

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

`rArr y=0" is the asymptote "`

Oblique asymptotes occur when the degree of the numerator > degree of the denominator. This is not the case here hence there are no oblique asymptotes.
graph{1/(x-2) [-10, 10, -5, 5]}