# little-o

Suppose f(*x*) and g(*x*) are real valued functions defined for all *x* > *x*_{0} (where *x*_{0} 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:
10000*x* = o(*x*^{2}), log(*x*) =
o(*x*), and *x*^{n} =
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.

**See Also:** BigOh, SameOrderofMagnitude, AsymptoticallyEqual

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