Definition:Big-O Notation/Also defined as

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Also defined as

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

On $\mathsf{Pr} \infty \mathsf{fWiki}$, this is known as a uniform big-$\OO$ estimate.


The statement $f = \map \OO g$ is sometimes seen to be defined as:

$\ds \exists \alpha \in \R_{\ge 0}: \lim_{x \mathop \to \infty} \frac {\map f x} {\map g x} = \alpha$

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