Definition:O Notation/Big-O Notation/Also defined as

Also defined as
Some authors require that the inequality be valid on the entire domain of definition.

On, this is known as a uniform big-$\OO$ estimate.

The statement $f = \map \OO g$ is sometimes seen to be defined as:
 * $\ds \exists \alpha \in \R: \alpha \ge 0 : \lim_{x \mathop \to \infty} \frac {\map f x} {\map g x} = \alpha$

However, requiring that the limit exists is generally viewed to be too restrictive.