Suppose f(x) and g(x) are real valued functions defined for all x greater than some fixed positive real x0. We write
f(x) = O(g(x)) (and we say "f(x) is big-O of g(x)"),
if there is some constant C such that
|f(x)| < C.g(x)
That is, f(x) is O(g(x)) if f is bounded by a constant times g.
For example, 53x2+23x+500 = O(x2), sin(x) = O(1), and any polynomial in x of degree at most n is O(xn).
This big-O notation was introduced by P. Brachmann in 1894.