Is this an abuse of notation? (Limits) This exercise is from my textbook:
Let $I\subseteq \mathbb{R}$ be an open interval, let $f : I\rightarrow \mathbb{R}$ be differentiable on $I$, and suppose $f''(a)$ exists at $a\in I$. Show that:
$$f''(a) = \lim_{h\to 0} \frac{f(a+h)  - 2f(a) + f(a-h)}{h^2}$$
which I solved using L'Hospital's rule, but I wasn't satisfied with that so I started to looking for an alternative proof and I found the following which uses the definition of derivative:
\begin{equation*}
\begin{split}
    f''(a) &= \lim_{h\to 0} \frac{f'(a) - f'(a-h)}{h}\\
\\
   &= \lim_{h\to 0} \frac{\lim_{h\to 0}\frac{f(a+h)-f(a)}{h} - \lim_{h\to 0}\frac{f(a)-f(a-h)}{h}}{h}\\
\\
  &= \lim_{h\to 0}\frac{\frac{f(a+h)-f(a)}{h} - \frac{f(a)-f(a-h)}{h}}{h}\\
\\
 &= \lim_{h\to 0} \frac{f(a+h)  - 2f(a) + f(a-h)}{h^2}
\end{split}
\end{equation*}
Is this valid? It looks more like an abuse of notation.
 A: No, you haven't justified any of the steps. In your chain of four equalities, I'd say the second is an abuse of notation (absolutely terrible abuse of notation; never overuse dummy variables in several spots, That's one of the easiest ways to get confused and give incorrect answers/justifications) while the third is completely unjustified.
If all you do is apply the definitions (and some minor theorems), then this is what you should end up with:
\begin{align}
f''(a)&=\lim_{h\to 0}\frac{f'(a)-f'(a-h)}{h}\\
&=\lim_{h\to 0}\frac{\left(\lim\limits_{k\to 0}\frac{f(a+k)-f(a)}{k}\right) - \left(\lim\limits_{z\to 0}\frac{f(a-h+z)-f(a-h)}{z}\right)}{h}.
\end{align}
Each occurrence of a derivative has a limit of difference quotients. Now, you have to justify why we are allowed to use one and the same variable $h$ in all three cases and why we can dispense with the inner two limits. This is not a trivial matter at all. You have to use some further techniques, all involving some form of the mean-value theorem (which tells us how the derivative of a function is related to differences in the original function) in the guise of L'Hopital's rule or Taylor's theorem.
