Well, correct me if I am mistaken, but it's just needed to prove that such a function c exists, and no more. And the mean value theorem does that, right?
I've a question concerning spivak's proof of L'Hopital's rule (in chapter 11).
It goes like this,
lim (x tends to a) of f = 0
lim (x tends to a) of g = 0
lim (x tends to a) of f'/g' exists,
then,
lim (x tends to a) f/g exists and is equal to lim (x tends to a) of f'/g'
He...
Is this the right negation of a finite limit?
\neg(\lim_{x\to a}f(x)=L) \iff \exists \epsilon>0 \forall \delta>0\exists x: 0<|x-a|<\delta \Rightarrow |f(x)-L|\geq\epsilon \vee \neg\exists f(x)
Thanks.