I'm studying the $\lim_{x\to 0}\frac1x\cos(\frac1x)$. Can I understand that the limit does not exist simply splitting it into two parts? That is, $$\lim_{x\to 0}\frac1x\cos(\frac1x)=\lim_{x\to 0}\frac {\cos(\frac1x)}{x}.$$ Now, the original limit does not exist because the $\lim_{x\to 0}\cos(\frac1x)$ does not exist. I am using a quotient rule.
Another example could be the $\lim_{n\to\infty}\sqrt n\cos(nx)$ because I can split it in the following way $$\lim_{n\to\infty}\frac{\cos(nx)}{1/\sqrt n}.$$
Question. Is my little argument correct? Should I use something else (i.e another technique) to study this limit in order to understand if it does exist or not?
Thank you.