O notation

Little-o notation

For real or complex-valued functions of a real variable ${\displaystyle x}$ with ${\displaystyle g(x)>0}$ for sufficiently large${\displaystyle x}$, one writes $${\displaystyle f(x)=o(g(x))\quad {\text{ as }}x\to \infty }$$ if $${\displaystyle \lim _{x\to \infty }{\frac {f(x)}{g(x)}}=0.}$$ That is, for every positive constant ε there exists a constant ${\displaystyle x_{0}}$ such that $${\displaystyle |f(x)|\leq \varepsilon g(x)\quad {\text{ for all }}x\geq x_{0}.}$$ Intuitively, this means that ${\displaystyle g(x)}$ grows much faster than ${\displaystyle f(x)}$, or equivalently ${\displaystyle f(x)}$ grows much slower than ${\displaystyle g(x)}$. For example, one has