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:
 |
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.
Infinitely-small function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Infinitely-small_function&oldid=29188