Definition:O Notation

Given two functions $$f \ $$ and $$g \ $$, the statement

$$f = O (g) \ $$

is equivalent to the statement

$$\exists \alpha>0 : \lim_{x \to \infty} \frac{f(x)}{g(x)} = \alpha \ $$.

The statement

$$f = o(g) \ $$

is equivalent to the statement

$$\lim_{x\to \infty} \frac{f(x)}{g(x)} = 0 \ $$

These are called big "oh" and little "oh", respectively.

Definition of Big-O
Big-O notation is a one type of order notation for typically comparing 'run-times' or growth rates between two growth functions.

$$Suppose: f,g$$ are two functions.

$$f(n){\in}O(g(n))$$ iff, $${\exists}c>0,k{\geq}0$$, such that $${\forall}n>k, f(n){\leq}cg(n)$$

This is read as "f(n) is big oh of g(n)".