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(miliotsioner), 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) $ g(x) <= f(x) <= h(x). So if there is some point 'a' in that interval that lim(g(x))=L x->a and lim(h(x))=L x->a THEN lim(f(x))=L x->a too.
