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

1 Answer
May 1, 2016

vertical asymptote x = 2
horizontal asymptote y = 0


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]}