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?

I would say the limit "tends to" $\infty$ and therefore your sequence diverges.
When working out limits, what you're doing is looking at what happens as you get very close to some point. So when working out the limit as $n \to \infty$, you're looking to see what happens as $n$ gets very big and as you're not working (whatever) out at a specific point, I wouldn't say its correct to say it "equals". "Tends to" means that it gets close to that point.
