Question

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)`?

Answer 1

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`