Definition of $\liminf_{x\to+\infty}f(x)$

Question

Let $f:\mathbb R\to\mathbb R.$ Can someone give me the definition of$$\liminf_{x\to+\infty}f(x)\:\:\text{and}\;\;\limsup_{x\to+\infty}f(x)$$ and a reference in which I can find the properties of $\liminf_{x\to+\infty}f(x)$ && $\limsup_{x\to+\infty}f(x)$

Thanks

Answer 1

$$ \liminf_{x \to \infty} f(x) = \lim_{y \to \infty} \inf_{x \geq y} f(x), \limsup_{x \to \infty} f(x) = \lim_{y \to \infty} \sup_{x \geq y} f(x) $$ See Wikipedia. References for $\liminf$ and $\limsup$ can be found on any introductory analysis book.