Definition:Big-O Notation/Real/Infinity

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

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

is equivalent to:
 * $\exists c \in \R: c \ge 0: \exists x_0 \in \R : \forall x \in \R : \left({x \ge x_0 \implies \left\vert{f \left({x}\right)}\right\vert \le c \cdot \left\vert{g \left({x}\right)}\right\vert}\right)$

That is:
 * $\left\vert{f \left({x}\right)}\right\vert \le c \cdot \left\vert{g \left({x}\right)}\right\vert$

for $x$ sufficiently large.

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

Also defined as
Some authors require that $g \left({x}\right)$ be nonzero for $x$ sufficiently large.