TITLE: Analysis: Spotting why Proof/Derivation is Wrong (part in which proof is wrong)
QUESTION [7 upvotes]: I'm having trouble figuring out what the error is in the above derivative.  I have no idea what is happening in the first line, how an application of the Mean Value Theorem gives such an expression to the right.
I just verified, some of the other parts and they seem to be okay as $\lim_{x \to 0} x\sin(1/x)$ is indeed $0$ so limit of $2\xi\sin(1/\xi)$ would also be $0$.  
So, honestly I'm not sure what the error is in the above derivation. I could really use some help.  Thank you.

REPLY [6 votes]: The identity via Mean Value Theorem holds for some specific $\xi\in(0,x)$ and not for all $\xi$. There is indeed a sequence $\{\xi_{n} \} $ such that $\xi_{n} \to 0^{+}$ and $\cos(1/\xi_{n})\to 0$ (for example $\xi_{n} =2/((2n+1)\pi)$). Not every sequence will behave in this manner but one such sequence exists. Thus you can't conclude $\lim_{\xi\to 0}\cos(1/\xi)=0$. Thus some care must be taken when using mean value theorem and applying limit.