I gave an incorrect proof here : How can evaluate $\lim_{x\to0}\frac{\sin(x^2+\frac{1}{x})-\sin\frac{1}{x}}{x}$
I am confused as when considering the mistakes in my proof it seems the limit cannot be $0$. The method must thus be completely wrong even more wrong than the comments suggest.
Yet I believe there should be a big O proof possible and even one similar to the one I posted. The "paradox" Im getting at is perhaps clearer understood when considering that my method/the correct method should also work if sine is replaced by another function that has a Taylor series at $0$ with all $a_n$ larger or equal to $0$. (and for which the limit should also be $0$ ofcourse).
Very confused. Please keep in mind that I want a proof based on big O and not on trig identities or l'hopital rule.