Say we have some function $f\in C^1$. I would like to somehow bound the error that is made when the limit is approximated using some arbirary $h$, that is to find $E(x,h)$ such that: $$|f'(x)-\frac{f(x+h)-f(x)}{h}|\leq E(x,h)$$ where $h$ is some specific number (i.e. $h=0.01$). I do not recall any helpful theorem but I'm really rusty on my analysis so please, bear with me if it's totally obvious.
2nd part
What if on the other hand I wanted to find $E'(x,h)$ which entails: $$|f'(x)-\frac{f(x+h)-f(x-h)}{2h}|\leq E'(x,h)$$ I remember that two-sided limit is more precise so is it the case that $|E(x,h)|\geq|E'(x,h)|$ ?