Definition:Big-O Notation/Real/Infinity

Definition
Let $f$ and $g$ be real-valued or complex-valued functions defined on a neighborhood of $+\infty$.

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

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

That is, $|f(x)| \leq c \cdot |g(x)|$ for $x$ sufficiently large.

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