Question

Solve: $\underset{x\to 1}{lim}\frac{\left|x^3-x\right|}{x-x^3}$

Answer 1

$\underset{x\to 1}{lim}\frac{|x^3-x|}{x^3-x}$

When x is slightly less 1, $|x^3-x|=x-x^3$

$\underset{x\to 1^{-}}{lim}\frac{x-x^3}{x^3-x}=-1$

$\underset{x\to 1^{+}}{lim}\frac{x^3-x}{x^3-x}=+1$

Limit does not exsist as right hand limit is not equal to left hand limit.