What are the asymptote(s) and hole(s), if any, of # f(x) =(1-x)^2/(x^2-1)+(1-x)^2/(x^2-2x+1)#?

1 Answer
Dec 29, 2015

Answer:

Asymptotes are # y = 2, x= 1 and x = -1#

Explanation:

The solutions are easier to find if the expression is first simplified.
#f(x) = (1-x)^2/((x-1)(x+1)) + (1-x)^2 / (x-1)^2#
#f(x) = (1-x)^2/(-(1-x)(x+1)) + (1-x)^2 /(- (1-x))^2#
#f(x) = -(1-x)/(x+1) +1#
#f(x) = (-(1-x) + (x + 1))/(x+1)#
#f(x) = (2x)/(x+1)#
# Lim f(x) x-> oo = 2# giving the horizontal asymptoteas #y = 2#

Going back to the original form of the equation, the denominator approaches zero as x approaches #1# or #-1# so therefore there are vertical asymptotes at #x= -1# and #x = 1#