Question

How do you find the limit of `(x^3 - x)/(x-1)^2` as x approaches 1?

Answer 1

`lim_(x->1-)[(x^3-x)/(x-1)^2]=-oo`
`lim_(x->1+)[(x^3-x)/(x-1)^2]=+oo`

Explanation:

`lim_(x->1)(x^3-x)/(x-1)^2=(x(x+1)(x-1))/(x-1)^2=(x^2+x)/(x-1)=2/0=oo`

However when `lim_(x->1-)[(x^3-x)/(x-1)^2]=-oo` and `lim_(x->1+)[(x^3-x)/(x-1)^2]=+oo`. This is because you can approach `1` at either side and this is important because it will determine the sign of `x-1`.