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