As an illustration, suppose that we are interested in the properties of a function *f*(*n*) as n becomes very large. If *f*(*n*) = *n*^{2} + 3*n*, then as n becomes very large, the term 3*n* becomes insignificant compared to *n*^{2}. The function *f*(*n*) is said to be "*asymptotically equivalent* to *n*^{2}, as *n* → ∞". This is often written symbolically as *f*(*n*) ~ *n*^{2}, which is read as "*f*(*n*) is asymptotic to *n*^{2}".

An example of an important asymptotic result is the prime number theorem. The theorem states that if π(*x*) is the number of prime numbers that are less than or equal to x, then

Formally, given functions *f*(*x*) and *g*(*x*), we define a binary relation

if and only if (de Bruijn 1981, §1.4)

The symbol ~ is the tilde. The relation is an equivalence relation on the set of functions of x; the functions f and g are said to be *asymptotically equivalent*. The domain of f and g can be any set for which the limit is defined: e.g. real numbers, complex numbers, positive integers.

The same notation is also used for other ways of passing to a limit: e.g. *x* → 0, *x* ↓ 0, |*x*| → 0. The way of passing to the limit is often not stated explicitly, if it is clear from the context.

Although the above definition is common in the literature, it is problematic if *g*(*x*) is zero infinitely often as x goes to the limiting value. For that reason, some authors use an alternative definition. The alternative definition, in little-o notation, is that *f* ~ *g* if and only if

This page was last edited on 14 May 2018, at 04:45 (UTC).

Reference: https://en.wikipedia.org/wiki/Asymptotic_analysis under CC BY-SA license.

Reference: https://en.wikipedia.org/wiki/Asymptotic_analysis under CC BY-SA license.

- Mathematical Analysis
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