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A function of a variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050930/i0509301.png" /> whose absolute value becomes and remains larger than any given number as a result of variation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050930/i0509302.png" />. More exactly, a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050930/i0509303.png" /> defined in a neighbourhood of a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050930/i0509304.png" /> is called an infinitely-large function as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050930/i0509305.png" /> tends to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050930/i0509306.png" /> if for any number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050930/i0509307.png" /> it is possible to find a number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050930/i0509308.png" /> such that for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050930/i0509309.png" /> satisfying <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050930/i05093010.png" /> the inequality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050930/i05093011.png" /> holds. This fact may be written as follows:
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A function of a variable $x$ whose absolute value becomes and remains larger 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-large function as $x$ tends to $x_0$ if for any number $M>0$ it is possible to find a number $\delta=\delta(M)>0$ such that for all $x\neq x_0$ satisfying $|x-x_0|<\delta$ the inequality $|f(x)|>M$ holds. This fact may be written as follows:
  
<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/i050930/i05093012.png" /></td> </tr></table>
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$$\lim_{x\to x_0}f(x)=\infty.$$
  
 
The following are defined in a similar manner:
 
The following are defined in a similar manner:
  
<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/i050930/i05093013.png" /></td> </tr></table>
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$$\lim_{x\to x_0\pm0}f(x)=\pm\infty,$$
  
<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/i050930/i05093014.png" /></td> </tr></table>
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$$\lim_{x\to\pm\infty}f(x)=\pm\infty.$$
  
 
For example,
 
For example,
  
<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/i050930/i05093015.png" /></td> </tr></table>
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$$\lim_{x\to-\infty}f(x)=+\infty$$
  
means that for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050930/i05093016.png" /> it is possible to find a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050930/i05093017.png" /> such that the inequality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050930/i05093018.png" /> is valid for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050930/i05093019.png" />. The study of infinitely-large functions may be reduced to that of infinitely-small functions (cf. [[Infinitely-small function|Infinitely-small function]]), since <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050930/i05093020.png" /> will be infinitely small.
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means that for any $M>0$ it is possible to find a $\delta=\delta(M)>0$ such that the inequality $f(x)>M$ is valid for all $x<-\delta$. The study of infinitely-large functions may be reduced to that of infinitely-small functions (cf. [[Infinitely-small function|Infinitely-small function]]), since $\psi(x)=1/f(x)$ will be infinitely small.
  
  

Latest revision as of 12:21, 28 August 2014

A function of a variable $x$ whose absolute value becomes and remains larger 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-large function as $x$ tends to $x_0$ if for any number $M>0$ it is possible to find a number $\delta=\delta(M)>0$ such that for all $x\neq x_0$ satisfying $|x-x_0|<\delta$ the inequality $|f(x)|>M$ holds. This fact may be written as follows:

$$\lim_{x\to x_0}f(x)=\infty.$$

The following are defined in a similar manner:

$$\lim_{x\to x_0\pm0}f(x)=\pm\infty,$$

$$\lim_{x\to\pm\infty}f(x)=\pm\infty.$$

For example,

$$\lim_{x\to-\infty}f(x)=+\infty$$

means that for any $M>0$ it is possible to find a $\delta=\delta(M)>0$ such that the inequality $f(x)>M$ is valid for all $x<-\delta$. The study of infinitely-large functions may be reduced to that of infinitely-small functions (cf. Infinitely-small function), since $\psi(x)=1/f(x)$ will be infinitely small.


Comments

See also Infinitesimal calculus.

How to Cite This Entry:
Infinitely-large function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Infinitely-large_function&oldid=33180
This article was adapted from an original article by V.I. Bityutskov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article