I am having difficuly proving that the limit $$\lim_{x\to0}\frac{x}{|x|}$$ does not exist using the negation of the epsilon-delta definition. I have seen a similar question answered here but I am having trouble applying the same logic to the negation of the definition instead of finding a contradiction.
This is the definition I am trying to use: $$\forall L\in\mathbb{R},\exists\epsilon>0\text{ s.t }\forall\delta>0,\exists x\in\mathbb{R} \text{ s.t } 0<|x-a|<\delta\text{ and } |F(x)-L|\ge\epsilon$$