Notation

Common Name

Limit Test

$f\left(n\right)\in O\left(g\left(n\right)\right)$

Asymptotic Upper Bound

$\underset{x\to \infty }{\lim }\left|\frac{f\left(x\right)}{g\left(x\right)}\right|<\infty$

$f\left(n\right)\in o\left(g\left(n\right)\right)$

Asymptotically Negligable

$\underset{x\to \infty }{\lim }\left|\frac{f\left(x\right)}{g\left(x\right)}\right|=0$

$f\left(n\right)\in \Omega \left(g\left(n\right)\right)$

Asymptotic Lower Bound

$\underset{x\to \infty }{\lim }\left|\frac{f\left(x\right)}{g\left(x\right)}\right|>0$

$f\left(n\right)\in \omega \left(g\left(n\right)\right)$

Asymptotically Dominant

$\underset{x\to \infty }{\lim }\left|\frac{f\left(x\right)}{g\left(x\right)}\right|=\infty$

$f\left(n\right)\in \Theta \left(g\left(n\right)\right)$

Asymptotically Tight Bound

$0<\underset{x\to \infty }{\lim }\left|\frac{f\left(x\right)}{g\left(x\right)}\right|<\infty$