Little-o notation

Little-o notation is a notation representing the behavior of a limit of a function at a given value. The statement

$f(x)=o(g(x))\,\,\,{\text{as}}\,\,\,x\to a$

can be intuitively interpreted as saying that g(x) grows much faster than f(x) at a, or, more mathematically,

$\lim _{x\to a}{\frac {f(x)}{g(x)}}=0$

Or, in cases where there is a third function, h(x):

$f(x)=h(x)+o(g(x))\Rightarrow \lim _{x\to a}{\frac {f(x)-h(x)}{g(x)}}=0$

For example:

• $x=o(x^{2})\,\,\forall x$
• $x^{2}\neq o(x^{2})\,\,\forall x$
• $\sin(x)\neq o(x),x\rightarrow 0$
• $\sin(x)\neq o(x^{2}),x\rightarrow 0$
• ${\sqrt {x}}={\sqrt {3}}+o(1),x\rightarrow 3$