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A function of a variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050940/i0509401.png" /> whose absolute value becomes and remains smaller than any given number as a result of variation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050940/i0509402.png" />. More exactly, a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050940/i0509403.png" /> defined in a neighbourhood of a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050940/i0509404.png" /> is called an infinitely-small function as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050940/i0509405.png" /> tends to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050940/i0509406.png" /> if for any number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050940/i0509407.png" /> it is possible to find a number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050940/i0509408.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050940/i0509409.png" /> is true for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050940/i05094010.png" /> satisfying the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050940/i05094011.png" />. This fact can be written as follows:
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{{TEX|done}}
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050940/i05094012.png" /></td> </tr></table>
 
  
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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:
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\begin{equation}
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\lim_{x\to x_0}f(x)=0.
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\end{equation}
 
Further, the symbolic notation
 
Further, the symbolic notation
 
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\begin{equation}
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050940/i05094013.png" /></td> </tr></table>
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\lim_{x\to+\infty}f(x)=0
 
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\end{equation}
means that for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050940/i05094014.png" /> it is possible to find an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050940/i05094015.png" /> such that for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050940/i05094016.png" /> the inequality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050940/i05094017.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050940/i05094018.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050940/i05094019.png" /> is finite and equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050940/i05094020.png" /> if and only if
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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
 
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\begin{equation}
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050940/i05094021.png" /></td> </tr></table>
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\lim_{x\to x_0} f(x)-A=0,
 
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\end{equation}
i.e. if the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050940/i05094022.png" /> is infinitely small. See also [[Infinitesimal calculus|Infinitesimal calculus]].
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i.e. if the function $f(x)-A$ is infinitely small. See also [[Infinitesimal calculus|Infinitesimal calculus]].

Latest revision as of 09:21, 14 December 2012


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=15375
This article was adapted from an original article by V.I. Bityutskov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article