definition of limsup of a function I already have some idea on what $\limsup_{x\rightarrow\infty} f(x)=\infty$ means but I would like to hear other ideas from the math stack exchange community. Maybe some intuition would be helpful. 

We define $\limsup_{x\to \infty}f(x)$ to be the maximum limit of any sequence $f(x_n)$ where $x_n \to \infty$.
More precisely, define $S:= \{(x_n): x_n \to \infty\}$, and $L:=\{\lim f(x_n): (x_n) \in S, \text{ and } \lim f(x_n) \text{ exists } \}$.
Then $\limsup_{x \to \infty}f(x):= \max L$. It is not immediate that this maximum exists, but it doesn't take too much work to show this.
