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A limit of a function of several variables in which the passage to the limit is performed successively in the different variables. Let, for example, a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081290/r0812901.png" /> of two variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081290/r0812902.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081290/r0812903.png" /> be defined on a set of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081290/r0812904.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081290/r0812905.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081290/r0812906.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081290/r0812907.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081290/r0812908.png" /> be limit points of the sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081290/r0812909.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081290/r08129010.png" />, respectively, or the symbol <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081290/r08129011.png" /> (if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081290/r08129012.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081290/r08129013.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081290/r08129014.png" /> and, respectively, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081290/r08129015.png" /> may be infinities with signs: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081290/r08129016.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081290/r08129017.png" />). If for any fixed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081290/r08129018.png" /> the limit
+
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$#A+1 = 42 n = 0
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$#C+1 = 42 : ~/encyclopedia/old_files/data/R081/R.0801290 Repeated limit
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<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/r/r081/r081290/r08129019.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
{{TEX|auto}}
 +
{{TEX|done}}
  
exists, and for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081290/r08129020.png" /> the limit
+
A limit of a function of several variables in which the passage to the limit is performed successively in the different variables. Let, for example, a function  $  f $
 +
of two variables  $  x $
 +
and  $  y $
 +
be defined on a set of the form  $  X \times Y $,
 +
$  x \in X \subset  \mathbf R  ^ {m} $,
 +
$  y \in Y \subset  \mathbf R  ^ {n} $,  
 +
and let  $  x _ {0} $
 +
and  $  y _ {0} $
 +
be limit points of the sets  $  X $
 +
and $  Y $,
 +
respectively, or the symbol  $  \infty $(
 +
if  $  m = 1 $
 +
or  $  n = 1 $,
 +
$  x _ {0} $
 +
and, respectively,  $  y _ {0} $
 +
may be infinities with signs: $  + \infty $,
 +
$  - \infty $).  
 +
If for any fixed  $  y \in Y $
 +
the limit
  
<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/r/r081/r081290/r08129021.png" /></td> </tr></table>
+
$$ \tag{1 }
 +
\phi ( y)  = \lim\limits _ {x \rightarrow x _ {0} }  f( x, y)
 +
$$
 +
 
 +
exists, and for  $  \phi ( y) $
 +
the limit
 +
 
 +
$$
 +
\lim\limits _ {y \rightarrow y _ {0} }  \phi ( y)
 +
$$
  
 
exists, then this limit is called the repeated limit
 
exists, then this limit is called the repeated limit
  
<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/r/r081/r081290/r08129022.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
$$ \tag{2 }
 +
\lim\limits _ {y \rightarrow y _ {0} }  \lim\limits _ {x \rightarrow x _ {0} }  f( x, y)
 +
$$
  
of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081290/r08129023.png" /> at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081290/r08129024.png" />. Similarly one defines the repeated limit
+
of the function $  f $
 +
at the point $  ( x _ {0} , y _ {0} ) $.  
 +
Similarly one defines the repeated limit
  
<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/r/r081/r081290/r08129025.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>
+
$$ \tag{3 }
 +
\lim\limits _ {x \rightarrow x _ {0} }  \lim\limits _ {y \rightarrow y _ {0} }  f( x, y).
 +
$$
  
 
If the (finite or infinite) [[Double limit|double limit]]
 
If the (finite or infinite) [[Double limit|double limit]]
  
<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/r/r081/r081290/r08129026.png" /></td> <td valign="top" style="width:5%;text-align:right;">(4)</td></tr></table>
+
$$ \tag{4 }
 +
\lim\limits _ {( x, y) \rightarrow ( x _ {0} , y _ {0} ) }  f( x, y)
 +
$$
  
exists, and if for any fixed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081290/r08129027.png" /> the finite limit (1) exists, then the repeated limit (2) also exists, and it is equal to the double limit (4).
+
exists, and if for any fixed $  y \in Y $
 +
the finite limit (1) exists, then the repeated limit (2) also exists, and it is equal to the double limit (4).
  
If for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081290/r08129028.png" /> the finite limit (1) exists, for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081290/r08129029.png" /> the finite limit
+
If for each $  y \in Y $
 +
the finite limit (1) exists, for each $  x \in X $
 +
the finite limit
  
<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/r/r081/r081290/r08129030.png" /></td> </tr></table>
+
$$
 +
\psi ( x)  = \lim\limits _ {y \rightarrow y _ {0} }  f( x, y)
 +
$$
  
exists, and for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081290/r08129031.png" /> the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081290/r08129032.png" /> tends to a limit function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081290/r08129033.png" /> uniformly in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081290/r08129034.png" />, then both the repeated limits (2) and (3) exist and are equal to one another.
+
exists, and for $  x \rightarrow x _ {0} $
 +
the function $  f( x, y) $
 +
tends to a limit function $  \phi ( y) $
 +
uniformly in $  y \in Y $,  
 +
then both the repeated limits (2) and (3) exist and are equal to one another.
  
If the sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081290/r08129035.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081290/r08129036.png" /> are sets of integers, then the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081290/r08129037.png" /> is called a double sequence, and the values of the argument are written as subscripts:
+
If the sets $  X $
 +
and $  Y $
 +
are sets of integers, then the function $  f $
 +
is called a double sequence, and the values of the argument are written as subscripts:
  
<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/r/r081/r081290/r08129038.png" /></td> </tr></table>
+
$$
 +
f( m, n)  = u _ {mn} ,
 +
$$
  
 
and the repeated limits
 
and the repeated limits
  
<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/r/r081/r081290/r08129039.png" /></td> </tr></table>
+
$$
 +
\lim\limits _ {n \rightarrow \infty }  \lim\limits _ {m \rightarrow \infty }  u _ {mn} \  \textrm{ and } \  \lim\limits _ {m \rightarrow \infty }  \lim\limits _ {n \rightarrow \infty }  u _ {mn}  $$
  
 
are called the repeated limits of the double sequence.
 
are called the repeated limits of the double sequence.
  
The concept of a repeated limit has been generalized to the case where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081290/r08129040.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081290/r08129041.png" /> and the set of values of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081290/r08129042.png" /> are subsets of certain topological spaces.
+
The concept of a repeated limit has been generalized to the case where $  X $
 +
and $  Y $
 +
and the set of values of the function $  f $
 +
are subsets of certain topological spaces.

Latest revision as of 08:11, 6 June 2020


A limit of a function of several variables in which the passage to the limit is performed successively in the different variables. Let, for example, a function $ f $ of two variables $ x $ and $ y $ be defined on a set of the form $ X \times Y $, $ x \in X \subset \mathbf R ^ {m} $, $ y \in Y \subset \mathbf R ^ {n} $, and let $ x _ {0} $ and $ y _ {0} $ be limit points of the sets $ X $ and $ Y $, respectively, or the symbol $ \infty $( if $ m = 1 $ or $ n = 1 $, $ x _ {0} $ and, respectively, $ y _ {0} $ may be infinities with signs: $ + \infty $, $ - \infty $). If for any fixed $ y \in Y $ the limit

$$ \tag{1 } \phi ( y) = \lim\limits _ {x \rightarrow x _ {0} } f( x, y) $$

exists, and for $ \phi ( y) $ the limit

$$ \lim\limits _ {y \rightarrow y _ {0} } \phi ( y) $$

exists, then this limit is called the repeated limit

$$ \tag{2 } \lim\limits _ {y \rightarrow y _ {0} } \lim\limits _ {x \rightarrow x _ {0} } f( x, y) $$

of the function $ f $ at the point $ ( x _ {0} , y _ {0} ) $. Similarly one defines the repeated limit

$$ \tag{3 } \lim\limits _ {x \rightarrow x _ {0} } \lim\limits _ {y \rightarrow y _ {0} } f( x, y). $$

If the (finite or infinite) double limit

$$ \tag{4 } \lim\limits _ {( x, y) \rightarrow ( x _ {0} , y _ {0} ) } f( x, y) $$

exists, and if for any fixed $ y \in Y $ the finite limit (1) exists, then the repeated limit (2) also exists, and it is equal to the double limit (4).

If for each $ y \in Y $ the finite limit (1) exists, for each $ x \in X $ the finite limit

$$ \psi ( x) = \lim\limits _ {y \rightarrow y _ {0} } f( x, y) $$

exists, and for $ x \rightarrow x _ {0} $ the function $ f( x, y) $ tends to a limit function $ \phi ( y) $ uniformly in $ y \in Y $, then both the repeated limits (2) and (3) exist and are equal to one another.

If the sets $ X $ and $ Y $ are sets of integers, then the function $ f $ is called a double sequence, and the values of the argument are written as subscripts:

$$ f( m, n) = u _ {mn} , $$

and the repeated limits

$$ \lim\limits _ {n \rightarrow \infty } \lim\limits _ {m \rightarrow \infty } u _ {mn} \ \textrm{ and } \ \lim\limits _ {m \rightarrow \infty } \lim\limits _ {n \rightarrow \infty } u _ {mn} $$

are called the repeated limits of the double sequence.

The concept of a repeated limit has been generalized to the case where $ X $ and $ Y $ and the set of values of the function $ f $ are subsets of certain topological spaces.

How to Cite This Entry:
Repeated limit. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Repeated_limit&oldid=17217
This article was adapted from an original article by L.D. Kudryavtsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article