Show that $\lim_{x\to |x_0|} \left(x\right) = |x_0|,\,$

Question

how do i make the correlation that |x-c| < d is equivalent to |x- $x_0$| < d.

Is it as simple as outright stating it or do I need to prove the question via contradiction.

Answer 1

Given $\epsilon >0$ you have to find $\delta >0$ such that $| |x| - |x_0| | <\epsilon$ whenever $|x-x_0|<\delta$.

Hint: Prove that, for all $x,y\in\mathbb{R}$ $$ ||x|-|y|| \leq |x-y|.$$