Understanding when it's OK to use limits arithmetic of multiplying

You surely can apply $$ \lim_{n\to\infty}a_nb_n= \Bigl(\lim_{n\to\infty}a_n\Bigr) \Bigl(\lim_{n\to\infty}b_n\Bigr) \tag{*} $$ when both limits on the right-hand side exist and are finite.

If one of the limit does not exist, you cannot even write (*) to begin with, so it doesn't make sense to ask whether you can apply it.

However, it's possible to add “rules” when one of those limits is infinity; essentially, if $\lim_{n\to\infty}b_n=c>0$ and $\lim_{n\to\infty}a_n=\infty$, also $\lim_{n\to\infty}a_nb_n=\infty$. Similar (and obvious) rules hold when $c<0$ or the other limit is $-\infty$.

The proof for the extended rule stated above is easy. Since $\lim_{n\to\infty}b_n=c>0$, there exists $N_0$ such that, for $n>N_0$, $$ b_n>\frac{c}{2} $$ Now take $K>0$; since $\lim_{n\to\infty}a_n=\infty$, there exists $N$ such that $N>N_0$ and, for all $n>N$, $a_n>2K/c$. Then, for $n>N$, $$ a_nb_n>\frac{2K}{c}\frac{c}{2}=K $$

In your case, $$ \lim_{n\to\infty}n=\infty \qquad \lim_{n\to\infty}\cos\frac{1}{n}=1 $$ and therefore $$ \lim_{n\to\infty}n\cos\frac{1}{n}=\infty $$

Note, however, that no rule can be stated when $\lim_{n\to\infty}a_n=\infty$ and $\lim_{n\to\infty}b_n=0$.