TITLE: Why does L'Hôpital's rule work?
QUESTION [35 upvotes]: Can anyone tell me why the L'Hôpital's rule works for evaluating limits of the form $\frac{0}{0}$ and $\frac{\infty}{\infty}$ ? 
What I understand about limits is that when you divide a really small thing (that is $\rightarrow0$) by another really small thing, we get a finite value which may not be so small. 
So how does differentiating the numerator and denominator help us get the Limit of a function?

REPLY [4 votes]: (http://csclub.uwaterloo.ca/~jy2wong/jenerated/blog/2012-10-07.lhopitals_rule.html) is interesting. 

At the heart of it though, L'Hopital's rule just seems to be a marriage of the ideas that differentiable functions are pretty darn close to their linear approximations at some point as long as you don't stray too far from that point and that for a continuous function, a small movement in the domain means a small movement in the value of the function.