differentiability and continuity

(a) Suppose that is differentiable and, that as . Prove that, if as , then .

(b) Give an example of a function such that as , but does not tend to a limit as

(a) I know f(x) must increase slower that x. So for some , , . Now the trouble of showing that this inequality increases until . Dunno what to do!

(b) I tried all the functions I could think of. I figured it should be oscillating since we dont want the derivative to converge... But it doesnt seem to make since since a converged function/series isnt increasing or decreasing anymore. I thought of something like but the derivative is complex. Otherwise I was trying things like sin(1/x) but couldnt find anything that worked.