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Infinitely-small function

From Encyclopedia of Mathematics
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A function of a variable whose absolute value becomes and remains smaller than any given number as a result of variation of . More exactly, a function defined in a neighbourhood of a point is called an infinitely-small function as tends to if for any number it is possible to find a number such that is true for all satisfying the condition . This fact can be written as follows:

Further, the symbolic notation

means that for any it is possible to find an such that for all the inequality is true. The concept of an infinitely-small function may serve as a base of the general definition of the limit of a function. In fact, the limit of the function as is finite and equal to if and only if

i.e. if the function is infinitely small. See also Infinitesimal calculus.

How to Cite This Entry:
Infinitely-small function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Infinitely-small_function&oldid=29188
This article was adapted from an original article by V.I. Bityutskov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article