I conjecture that in some specific conditions a differentiating function gives the following equality: $\lim_{n\to\infty} f(n+1) - f(n) = \lim_{x\to\infty} (f(x))' $
However, I'm not sure yet what exactly those conditions are in order to precisely know where
I may apply this rule or not. If you wanna take a look over my posted problem here you'll immediately notice that this rule applies for that case. I really appreciate if you help me clarify this.