I have always had a little trouble fully understanding iterated limits. So I thought it might be a good exercise to expand it in quantifiers.
Suppose we want to expand $\displaystyle \lim_{x \to a} f'(x) = L$. So we want to expand
$$\lim_{x \to a} \left( \lim_{h \to x} \dfrac {f(h) - f(x)}{h-x} \right) = L$$
So first expanding the outer limit (I'm not even sure how to expand the inner limit first) we have
$$\forall (\epsilon > 0) \exists (\delta > 0) \forall(x \in \mathbb{R}): |x-a| < \delta \implies \left| \lim_{h \to x} \dfrac {f(h) - f(x)}{h-x} -L \right| < \epsilon $$
However, I'm not really sure how to expand this further.
Any help in this regard, or in understanding iterated limits better, is greatly appreciated. Thank you.