Informally, the term asymptotic means approaching a value or curve arbitrarily closely (i.e., as some sort of limit is taken). A line or curve A that is asymptotic to given curve C is called the asymptote of C.

More formally, let x be a continuous variable tending to some limit. Then a real function f(x) and positive function phi(x) are said to be asymptotically equivalent, written f∼phi, if


as the limit is taken.

Equivalently, consider the little-o asymptotic notation o(x) that is one of the Landau symbols. Then f=o(phi) means that


as a limit is taken. The statement f∼phi is then equivalent to




(Hardy and Wright 1979, pp. 7-8).

These definitions can also be applied to the discrete case of n an integer variable that tends to infinity, f(n) a real function of n, and phi(n) a positive function of n.

See also

Asymptosy, Asymptote, Asymptotic Curve, Asymptotic Direction, Asymptotic Notation, Asymptotic Series, Big-O Notation, Landau Symbols, Limit, Little-O Notation, Order of Magnitude

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Hardy, G. H. and Wright, E. M. An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon Press, pp. 7-8, 1979.

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Cite this as:

Weisstein, Eric W. "Asymptotic." From MathWorld--A Wolfram Web Resource.

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