Notation in regards to limits Do we define infinity as a "limit"? Or do we simply say that the limit doesn't exist as the function/series diverges? 

I'm calculating the limit of a function, turns out to be infinity, but I am not sure whether to just write "infinity" after an equal sign, or to say "this doesn't exist". 

Also, can I even use the term: "Lim f(x) = " without knowing whether the limit exists beforehand?
you can say the limit is positively (or negatively) divergent and it would be correct, but among mathematicians no one looks bad on:
$$\lim_{x\to x_0}f(x)=+\infty$$
because the result of a limit is not necessarely in $\mathbb{R}$, most analists  use this definition: $\mathbb{\hat R}=\mathbb{R}\  \cup \{-\infty,+\infty\}$ (the extended real line) and if you assume that the result lies in $\mathbb{\hat R}$ then it's perfectly rigorous to write the limit result using the notation above. (for more info look: here)
$\infty$ is not a number. 

Consider for example if limit of some function tends to infinity at some point c 

than it means that if you get closer and closer to that point than the function just doesn't have bounds.

We say that the limit exists but it is not bounded which is indicated as $\infty$.

i.e.,

if you get closer and closer towards c then your limit doesn't have any bounds.