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.