Given a differentiable function $f$, prove that $\lim_{h\to 0}\frac{f\left(x + \frac{h}{2}\right)-f\left(x - \frac{h}{2}\right)}{h} = f'(x)$.

Question

I'm trying to prove the following statement

Given a differentiable function $f:\mathbb{R}\to \mathbb{R}$, prove that $$\lim_{h \to 0} \frac{f\left(x + \frac{h}{2}\right)-f\left(x - \frac{h}{2}\right) }{h} = f'(x)$$

I know $2$ definitions for the derivative of a function $f$: $$ \lim_{h \to 0} \frac{f\left(x +h\right)-f\left(x\right) }{h} \qquad \text{and} \qquad \lim_{a \to x} \frac{f\left(x \right)-f\left(a\right) }{x-a} $$ so my idea was to try to arrange the limit in question into one of these $2$ forms.

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I proceeded to take $x^* = x - \frac{h}{2} $ to do a change of variable, in which case my limit would end up looking like $$\lim_{h \to 0} \frac{f\left(x^* +h\right)-f\left(x^*\right) }{h} = f'(x^*) $$ which would seem to imply that this is equal to $f'\left(x - \frac{h}{2}\right)\neq f'(x)$ using the first definition of the derivative. I know that since $h \to 0$, then $f'\left(x - \frac{h}{2}\right)$ and $f'(x)$ are the same thing, but since the paramater $h$ is included in the substitution I'm making I don't know how to account for this after I took the limit.

I think my problem is just a basic concept misconception, but I can't seem to find a way to rigorously justify the steps I need to take to transform the limit into one of the definitions that I know. Could anyone tell me what I'm doing wrong or how I could correctly structure this argument to make it rigorous? Thank you!

Answer 1

$$\lim_{h \to 0} \frac{f\left(x + \dfrac{h}{2}\right)-f\left(x - \dfrac{h}{2}\right) }{h}=\lim_{h \to 0} \dfrac{f\left(x + \dfrac{h}{2}\right)-f(x)+f(x)-f\left(x - \dfrac{h}{2}\right) }{h} \\=\lim_{h \to 0} \dfrac{f\left(x + \dfrac{h}{2}\right)-f(x)}{2\dfrac h2} +\lim_{h \to 0} \dfrac{f\left(x - \dfrac{h}{2}\right)-f(x) }{-2\dfrac h2}=2\frac{f'(x)}2.$$

Note that the reciprocal is not true, because $f(x)$ does not appear in the symmetric formula. So the equivalence is one-way.

Answer 2

\begin{align} & \frac{f\left(x + \frac{h}{2}\right)-f\left(x - \frac{h}{2}\right) }{h} \\[12pt] = {} & \frac 1 2\left( \frac{f\left(x + \frac{h}{2}\right) - f(x)}{h/2} + \frac{f(x) -f\left(x - \frac{h}{2}\right)}{h/2} \right) \\[8pt] \to {} & \frac 1 2\Big( f'(x) + f'(x)\Big) \quad \text{as } h\to0. \end{align}