differentiability and continuity
(a) Suppose that $\displaystyle f: \Re\rightarrow\Re$is differentiable and, that $\displaystyle f(x)/x\rightarrow 0$ as $\displaystyle x\rightarrow\infty$. Prove that, if $\displaystyle f'(x)\rightarrow k$ as $\displaystyle x\rightarrow\infty$, then $\displaystyle k=0$.
(b) Give an example of a function $\displaystyle g:\Re\rightarrow\Re$ such that $\displaystyle g(x)\rughtarrow 0$ as $\displaystyle x\rightarrow\infty$, but $\displaystyle g'(x)$ does not tend to a limit as $\displaystyle x\rightarrow\infty$
(a) I know f(x) must increase slower that x. So for some $\displaystyle N\in\Re$, $\displaystyle x_1,x_2\in\Re$, $\displaystyle |f(x_1)-f(x_2)|<|x_1-x_2|$. Now the trouble of showing that this inequality increases until $\displaystyle \frac{|f(x_1)-f(x_2)|}{x_1-x_2|}<\epsilon$. 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 $\displaystyle (-1)^x/x$ but the derivative is complex. Otherwise I was trying things like sin(1/x) but couldnt find anything that worked.