Im struggling with a couple of questions... any help would be appreciated. You have to answer true or false and if true give a proof, if false give a counterexample.

(a) If lim(f(x)) as x tends to infinity is finite and limf'(x)=b then b=0.

I think this is true but i have no idea how to prove it, I wouldn't even know where to start!

(b) If lim(f(x)) as x tends to infinity is finite, the lim(f'(x))=0.

I think this must be false because of the subtle difference in the question - is it false because the limit of the derivative function might not actually exist? This seems to make sense to me as the only way it could be false, although I cannot think of a counterexample.

thanks