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:
 * $\map f x = \map {\mathcal O} {\map g x}$ as $x \to \infty$

is equivalent to:
 * $\exists c \in \R: c \ge 0: \exists x_0 \in \R: \forall x \in \R: \paren {x \ge x_0 \implies \size {\map f x} \le c \cdot \size {\map g x} }$

That is:
 * $\size {\map f x} \le c \cdot \size {\map g x}$

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 $\map g x$ be nonzero for $x$ sufficiently large.