discrete differences, doubt about sign

Question

Let $G\in C(\mathbb R)$ then $\lim_{N\to +\infty}\frac{G(x-\frac{1}{N})-G(x)}{-\frac{1}{N}}=G'(x)$?

I have a doubt about the sign in front of $G'$, it is a $+$ or a $-$? Thanks to everyone

Answer 1

The definition of the derivative is usually stated as:

$$ G'(x) = \lim \limits_{h \rightarrow 0} \frac{G(x+h)-G(x)}{h} $$

If we replace $h$ by $-\frac{1}{N}$, we get the expression in your question. So, the sign in front of $G'$ is indeed positive.