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
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.