There is a given real function which is differentiable and such that
We're asked to give a proof that if $\;f\;$ is bounded, then the limit must be $\;L=0\;$
What I worked out: geometrically it looks very clear, but I'm having problems with the formal part.
I said: we suppose $\;|f(x)|\le M\;\;\;\forall\ x\in\Bbb R\;$ , and then I tried the mean value theorem. Since
there exists $\;c_x\in(o,x)\;$ for all positive $\;x\;$, so that
The problem I meet is that if I do$\;x\to\infty\;$ I cannot deduce that $\;c_x\to \infty\;$ because I don't know if we could be getting the same $\;c_x\;$ for different $\;x$'s. If I could solve this then I would get immediately that $\;L=0\;$ .
Any help will be appreciated.