definition of limsup of a function

Question

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.

Answer 1

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 $\infty$ or some real number if it does not exist (imagine a function going to infinity, or oscillating but getting slightly bigger with each oscillation, never reaching a maximum nor going to infinity). There are tons of other ways such a function can behave, but the supremum will always exist.

However, 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: $$\limsup_{x\to\infty} f(x) = \lim_{y\to\infty} \sup_{x\geq y} f(x).$$

Answer 2

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.