In my calculus class we are using the next definition of limit superior and inferior:

Let $(x_n)\in\mathbb{R}$ be a sequence. The limit superior of $(x_n)$ is the extended real number $$\overline{\lim_{n\to \infty}} x_n :=\lim_{n\to \infty}\left(\sup_{k\geq n}x_k\right).$$ The limit inferior of $(x_n)$ is the extended real number $$\varliminf_{n \to \infty} x_n:=\lim_{n\to \infty}\left(\inf_{k\geq n}x_k\right).$$

I don't fully understand those definitions and sometimes, I make mistakes when doing exercise that involve such concepts. Any help is appreciated

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    $\begingroup$ It would help us to help you if you gave a specific example of an exercise that is giving you problems. $\endgroup$ – Rob Arthan Sep 16 at 23:27

There are two common definitions of $\limsup x_n.$ My favored of the two is "$\limsup x_n$ is the largest subsequential limit of $x_n.$" There are problems with this. Why should there be a largest subsequential limit? The other definition is the one you gave. It makes total logical sense, and is elementary. But why should anyone care about it? Then there is the definition given by my old professor: It's the "largest number that the sequence $x_n$ visits infinitely often". That's cute, but rife with undefined terms coming out of the woodwork. There you have it: 3 descriptions/definitions trying to get at the same basic idea.

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  • $\begingroup$ Your old professor had a screw loose, if you are quoting him or her accurately: $\limsup_{n \to \infty}(1 - 1/n) = 1$, but $1 - 1/n$ never "visits" $1$. It's right if you replace "visits infinitely often" by "gets arbitrarily near to infinitely often". I don't understand your objections to the standard definition as given by the OP. $\endgroup$ – Rob Arthan Sep 16 at 23:28

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