Definition:Big-O Notation/Real/Infinity

Definition
Let $f$ and $g$ be real-valued or complex-valued functions on a subset of $\R$ containing all sufficiently large real numbers.

The statement:
 * $f(x) = \mathcal O \left({g(x)}\right)$ for $x\to\infty$

is equivalent to the statement:
 * $\displaystyle \exists c\in \R: c\ge 0 : \exists x_0\in\R : |f(x)|\leq c\cdot|g(x)|$ for all $x\geq x_0$

This statement is voiced $f$ is big-O of $g$ or simply $f$ is big-O $g$.