How are $\limsup_{x \to \infty}f(x)$ and $\liminf_{x \to \infty}f(x)$ defined?

Question

In the below wiki article, $\limsup$ and $\liminf$ are discussed for several objects, including sequences, functions and sets.

However, although the article gives definitions for $\limsup_{x \to a}f(x)$ and $\liminf_{x \to a}f(x)$, where $f$ is a real function, it doesn't give a definition for $\limsup_{x \to \infty}f(x)$ or $\liminf_{x \to \infty}f(x)$.

How are these defined, please?

https://en.wikipedia.org/wiki/Limit_superior_and_limit_inferior#Functions_from_metric_spaces_to_complete_lattices

Answer 1

$$\limsup\limits_{x\to\infty}f(x)=\lim\limits_{x\to\infty}\sup\{f(y)\,:\, y>x\}\\ \liminf\limits_{x\to\infty}f(x)=\lim\limits_{x\to\infty}\inf\{f(y)\,:\, y>x\}$$

Equivalently:

• $\limsup_{x\to\infty}f(x)$ is the extended real number $\alpha$ such that, for all $\beta<\alpha$, $f(x)$ is frequently larger than $\beta$ as $x\to\infty$, and, for all $\gamma>\alpha$, $f(x)$ is eventually smaller than $\gamma$ as $x\to\infty$;

• $\liminf_{x\to\infty}f(x)$ is the extended real number $\alpha$ such that, for all $\beta<\alpha$, $f(x)$ is eventually larger than $\beta$ as $x\to\infty$, and, for all $\gamma>\alpha$, $f(x)$ is frequently smaller than $\gamma$ as $x\to\infty$.

Answer 2

$\limsup\limits_{x\to\infty}f(x) = \lim\limits_{N \to \infty} \sup\{f(x): x > N\}$

within the extended line $[-\infty, +\infty]$ it is always defined, otherwise we need to require $f(x)$ be bounded when $x \to \infty$