Infinitely-small function
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:
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Further, the symbolic notation
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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
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i.e. if the function is infinitely small. See also Infinitesimal calculus.
Infinitely-small function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Infinitely-small_function&oldid=29188