Definition: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-O estimate.

The statement $f = \mathcal O \left({g}\right)$ is sometimes seen to be defined as:
 * $\displaystyle \exists \alpha \in \R: \alpha \ge 0 : \lim_{x \to \infty} \frac{f \left({x}\right)}{g \left({x}\right)} = \alpha$

But requiring that the limit exists is too restrictive.