Is it ever proper to say that the limit of a function equals infinity? If I calculate a limit and get the value $\infty$, what is the proper way to communicate this? Can I say that the $\lim_{n\to\infty}a_n=\infty$ and therefore the sequence $\{a_n\}$ diverges, or do I say that the limit does not exist?

Saying that the limit is infinity has the same precise meaning as writing $\lim_{n\to\infty}a_n=\infty$. Saying that the sequence diverges has not the same precise meaning, because the limit could also be $-\infty$.
I'd avoid phrases such as "the limit tends to"; the limit either doesn't exist or, if it exists, it is a real number, $\infty$ or $-\infty$.
In any case, it's a "local" question: terminology is not fixed; somebody uses "divergent" as synonymous to "non convergent", for instance. But what you're allowed to write you should be also allowed to say.
