little-o

Suppose f(x) and g(x) are real valued functions defined for all x > x₀ (where x₀ is a fixed positive real). We write

f(x) = o(g(x))

If the limit as x approaches infinity of f(x)/g(x) is zero (that is, if eventually f(x)/g(x) becomes less than any given positive number). Examples: 10000x = o(x²), log(x) = o(x), and xⁿ = o(e^x). Notice that f(x) = o(g(x)) implies, and is stronger than, f(x) = O(g(x)).

We often use the little-oh notation this way:

f(x) = g(x) + o(h(x)).

This intuitively means that the error in using g(x) to approximate f(x) is negligible in comparison to h(x).

The little-oh notation was first used by E. Landau in 1909.