How do you quantify:
A function $f:\mathrm{dom}(f) \longrightarrow \mathrm{codom}(f)$ is differentiable at every $x$ contained in $\mathrm{dom}(f)$ if the limit $\lim_{h \to 0}\frac{f(x+h)-f(x)}{h}$ exists.
I have looked everywhere for a quantified definition of the above and have only found quantified versions that don't utilize the limit as $h$ approaches zero. Here is my attempt:
$(\forall \varepsilon > 0)(\exists \delta > 0)(\forall x)\left( |h| < \delta \Rightarrow \left| f'(x) - \frac{f(x + h) - f(x)}{h} \right| < \varepsilon \right).$
I believe this to be correct however as I am teaching myself analysis, I am being extra cautious with everything.