TITLE: definition of limsup of a function
QUESTION [14 upvotes]: 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.

REPLY [24 votes]: If you are talking about $\limsup,$ I assume you have a good bit of familiarity with $\sup.$
As an example: the supremum of a function on the interval $[0,\infty)$ is just its maximum, if the maximum exists, and $\infty$ if it does not exist. The supremum doesn't tell you anything about how $f(x)$ behaves as $x\to\infty.$ This is the job of $\limsup.$ $\limsup$ tries to ignore what happens on any finite interval $[0,a]$ and tells you the supremum once you look further and further away. In other words, it is the limit of the supremums as we look further away: $$\limsup_{x\to\infty} f(x) = \lim_{y\to\infty} \sup_{x\geq y} f(x).$$