# Infinitely-small function

A function of a variable $x$ whose absolute value becomes and remains smaller than any given number as a result of variation of $x$. More exactly, a function $f$ defined in a neighbourhood of a point $x_0$ is called an infinitely-small function as $x$ tends to $x_0$ if for any number $\varepsilon>0$ it is possible to find a number $\delta=\delta(\varepsilon)>0$ such that $|f(x)|<\varepsilon$ is true for all $x$ satisfying the condition $|x-x_0|<\delta$. This fact can be written as follows:
\begin{equation}
\lim_{x\to x_0}f(x)=0.
\end{equation}
Further, the symbolic notation
\begin{equation}
\lim_{x\to+\infty}f(x)=0
\end{equation}
means that for any $\varepsilon>0$ it is possible to find an $N=N(\varepsilon)>0$ such that for all $x>N$ the inequality $|f(x)|<\varepsilon$ 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 $f$ as $x\to x_0$ is finite and equal to $A$ if and only if
\begin{equation}
\lim_{x\to x_0} f(x)-A=0,
\end{equation}
i.e. if the function $f(x)-A$ 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