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Asymptotic limit


In mathematical analysis, asymptotic analysis is a method of describing limiting behavior. The method has applications across science. Examples are:

A simple illustration, when considering a function f(n), is when there is a need to describe its properties as n becomes very large. Thus, if f(n) = n2+3n, the term 3n becomes insignificant compared to n2, when n is very large. The function f(n) is said to be "asymptotically equivalent to n2 as n → ∞", and this is written symbolically as f(n) ~ n2.

Formally, given functions f and g of a natural number variable n, one defines a binary relation

if and only if (according to Erdelyi, 1956)

This relation is an equivalence relation on the set of functions of n. The equivalence class of f informally consists of all functions g which are approximately equal to f in a relative sense, in the limit.

If , then

for any real r, and


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