TITLE: Does L'hopital work for one sided limits?
QUESTION [6 upvotes]: Simple question, to which I don't know the answer. Does it work the same even if we are only interested in one-sided limits, and it won't cause problems that the actual limit doesn't exist?

REPLY [3 votes]: One of the key assumptions of L'Hospital's theorem is that the functions $f,g$ be derivable over an open interval one of whose end points is $c$, the point to which $x$ tends:
$$\lim_{x\rightarrow c}\frac{f(x)}{g(x)}.$$
The proof due to Taylor and presented on Wiki does operate only over this one sided interval of common differentiability.
So the L'Hospital rule is basically about one sided limits. 
To be very rigorous, first, one has to examine the derivatives on both sides of $c$. It may turn out that the left and the right limits differ or only one of them exits.