calculus - Showing that $limlimits_{x→0}frac{xsin(1/x)-cos(1/x)}x$ does not exist without sequence

Show that $$\lim_{x→0}\frac{x\sin(1/x)-\cos(1/x)}x$$ does not exist.

I understand that at $0$, the $\dfrac{\cos(1/x)}x$ term varies between $(-\infty , + \infty)$. But I want a complete formal proof that use the definition of limit ($ε, δ$) and not using sequences like ($2k\pi n$) or other techniques like l'Hopital, etc… How to write a formal proof for that?

Also I want to show that $\lim\limits_{x\to0} (x\sin(1/x) - \cos(1/x))$ does not exist without using any sequences and only using the definition of limit.