One theorem that could be a good candidate for my favorite one is is mm.. there are many funny names to it: the theorem about two policemen(militsioner), the sandwich theorem, squeeze theorem.. So the definition. We have 3 functions f, g, h on some certain interval, and they hold following inequality on this interval: $ g(x)\le f(x) \le h(x) $. So if there is some point 'a' in that interval that $ \lim_{x \to a}g(x) = L $ and $ \lim_{x \to a}h(x) = L $ THEN $ \lim_{x \to a}f(x) = L $ too. So basically, g and h "squeeze" the f function into that L limit. Funny name and intuitively easy to understand theorem, but important one in solving and proving other theorems and problems.
