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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.

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    $\begingroup$ $\displaystyle\limsup_{x \to \infty} f(x) = \infty$ means that the values of $f(x)$ are unbounded from above on all neighborhoods of $+\infty.$ By a neighborhood of $+\infty,$ I mean an open interval of the form $(a, \infty)$ for some $a \in {\mathbb R}.$ In other words, no matter how far to the right on the number line you go and stand, the values of the function $f$ will be arbitrarily large (positively) for some of the values of $x$ lying to the right of where you're standing. $\endgroup$ – Dave L. Renfro Oct 22 '13 at 20:38
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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: $$\limsup_{x\to\infty} f(x) = \lim_{y\to\infty} \sup_{x\geq y} f(x).$$

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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.

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