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The double limit of a sequence, the limit of a double sequence, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033890/d0338901.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033890/d0338902.png" /> is a number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033890/d0338903.png" /> which is defined as follows: For any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033890/d0338904.png" /> there exists an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033890/d0338905.png" /> such that for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033890/d0338906.png" /> the equality
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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/d/d033/d033890/d0338907.png" /></td> </tr></table>
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 +
The double limit of a sequence, the limit of a double sequence,  $  \{ x _ {mn} \} $,
 +
$  m , n = 1 , 2 \dots $
 +
is a number  $  a $
 +
which is defined as follows: For any  $  \epsilon > 0 $
 +
there exists an  $  N _  \epsilon  $
 +
such that for all  $  m , n > N _  \epsilon  $
 +
the equality
 +
 
 +
$$
 +
| x _ {mn} - a |  < \epsilon
 +
$$
  
 
is valid. The notation is
 
is valid. The notation is
  
<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/d/d033/d033890/d0338908.png" /></td> </tr></table>
+
$$
 +
= \lim\limits _ {m , n \rightarrow \infty }  x _ {mn} .
 +
$$
  
If for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033890/d0338909.png" /> there is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033890/d03389010.png" /> such that for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033890/d03389011.png" /> the inequality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033890/d03389012.png" /> is fulfilled, then the sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033890/d03389013.png" /> has infinity as its limit:
+
If for any $  \epsilon > 0 $
 +
there is an $  N _  \epsilon  $
 +
such that for all $  m , n > N _  \epsilon  $
 +
the inequality $  | x _ {mn} | > \epsilon $
 +
is fulfilled, then the sequence $  x _ {mn} $
 +
has infinity as its 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/d/d033/d033890/d03389014.png" /></td> </tr></table>
+
$$
 +
\lim\limits _ {m , n \rightarrow \infty }  x _ {mn}  = \infty .
 +
$$
  
 
The infinite limits
 
The infinite 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/d/d033/d033890/d03389015.png" /></td> </tr></table>
+
$$
 +
\lim\limits _ {m,n \rightarrow \infty }  x _ {mn}  = + \infty \ 
 +
\textrm{ and } \ \
 +
\lim\limits _ {m,n \rightarrow \infty }  x _ {mn}  = - \infty
 +
$$
  
are defined in the same manner. The double limit of a sequence is a special case of the double limit of a function over a set, namely when this set consists of the points on the plane with integer coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033890/d03389016.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033890/d03389017.png" />. Accordingly, the double limit of a sequence is connected with its repeated limits as in the general case.
+
are defined in the same manner. The double limit of a sequence is a special case of the double limit of a function over a set, namely when this set consists of the points on the plane with integer coordinates $  m $
 +
and $  n $.  
 +
Accordingly, the double limit of a sequence is connected with its repeated limits as in the general case.
  
The double limit of a function is the limit of a function of two variables, defined as follows. Let the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033890/d03389018.png" /> be defined on a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033890/d03389019.png" /> in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033890/d03389020.png" />-plane, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033890/d03389021.png" /> be a limit point of it (cf. [[Limit point of a set|Limit point of a set]]). A number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033890/d03389022.png" /> is said to be the double limit of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033890/d03389023.png" /> at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033890/d03389024.png" />, or as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033890/d03389025.png" />, if for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033890/d03389026.png" /> there exists a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033890/d03389027.png" /> such that for all the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033890/d03389028.png" /> the coordinates of which satisfy the inequalities
+
The double limit of a function is the limit of a function of two variables, defined as follows. Let the function $  f ( x , y ) $
 +
be defined on a set $  E $
 +
in the $  X Y $-
 +
plane, and let $  ( x _ {0} , y _ {0} ) $
 +
be a limit point of it (cf. [[Limit point of a set|Limit point of a set]]). A number $  A $
 +
is said to be the double limit of the function $  f ( x , y ) $
 +
at the point $  ( x _ {0} , y _ {0} ) $,  
 +
or as $  ( x , y ) \rightarrow ( x _ {0} , y _ {0} ) $,  
 +
if for any $  \epsilon > 0 $
 +
there exists a $  \delta > 0 $
 +
such that for all the points $  ( x , y ) \in E $
 +
the coordinates of which satisfy the inequalities
  
<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/d/d033/d033890/d03389029.png" /></td> </tr></table>
+
$$
 +
< | x - x _ {0} |  < \delta ,\ \
 +
< | y - y _ {0} |  < \delta ,
 +
$$
  
 
the inequality
 
the inequality
  
<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/d/d033/d033890/d03389030.png" /></td> </tr></table>
+
$$
 +
| f ( x , y ) - A |  < \epsilon
 +
$$
  
 
is valid. In such a case the limit is written as
 
is valid. In such a case the limit is written as
  
<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/d/d033/d033890/d03389031.png" /></td> </tr></table>
+
$$
 +
\lim\limits _ {( x , y ) \rightarrow ( x _ {0} , y _ {0} ) } \
 +
f ( x , y )  = A .
 +
$$
  
 
The double limit of a function may be formulated in terms of the limit of a sequence:
 
The double limit of a function may be formulated in terms of the limit of a sequence:
  
<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/d/d033/d033890/d03389032.png" /></td> </tr></table>
+
$$
 +
= \lim\limits _ {( x, y ) \rightarrow ( x _ {0} , y _ {0} ) } \
 +
f ( x , y ) ,
 +
$$
  
 
if for any sequence
 
if for any sequence
  
<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/d/d033/d033890/d03389033.png" /></td> </tr></table>
+
$$
 +
( x _ {n} , y _ {n} )  \rightarrow  ( x _ {0} , y _ {0} ),
 +
$$
  
<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/d/d033/d033890/d03389034.png" /></td> </tr></table>
+
$$
 +
( x _ {0} , y _ {0} )  \neq  ( x _ {n} , y _ {n} )  \in  E ,\  n = 1, 2 \dots
 +
$$
  
 
the condition
 
the condition
  
<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/d/d033/d033890/d03389035.png" /></td> </tr></table>
+
$$
 +
\lim\limits _ {n \rightarrow \infty }  f ( x _ {n} , y _ {n} )  = A
 +
$$
  
is satisfied. The double limit of a function as its arguments tend to infinity, as well as the definitions of infinite double limits of a function, are formulated in a similar manner. There exists a connection between the double limit of a function and the [[Repeated limit|repeated limit]] of a function at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033890/d03389036.png" /> or at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033890/d03389037.png" />: Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033890/d03389038.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033890/d03389039.png" /> be limit points (finite or infinite) of the real subsets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033890/d03389040.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033890/d03389041.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033890/d03389042.png" />. If for a function a finite or infinite double limit
+
is satisfied. The double limit of a function as its arguments tend to infinity, as well as the definitions of infinite double limits of a function, are formulated in a similar manner. There exists a connection between the double limit of a function and the [[Repeated limit|repeated limit]] of a function at a point $  ( x _ {0} , y _ {0} ) $
 +
or at $  \infty $:  
 +
Let $  x _ {0} $
 +
and $  y _ {0} $
 +
be limit points (finite or infinite) of the real subsets $  X $
 +
and $  Y $,  
 +
and let $  E = X \times Y $.  
 +
If for a function a finite or infinite 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/d/d033/d033890/d03389043.png" /></td> </tr></table>
+
$$
 +
\lim\limits _ {( x , y ) \rightarrow ( x _ {0} , y _ {0} ) } \
 +
f ( x , y )
 +
$$
  
exists, and if for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033890/d03389044.png" /> there is a finite limit
+
exists, and if for any $  y \in Y $
 +
there is a 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/d/d033/d033890/d03389045.png" /></td> </tr></table>
+
$$
 +
\phi ( y)  = \lim\limits _ {x \rightarrow x _ {0} } \
 +
f ( x , y ) ,
 +
$$
  
 
then the repeated limit
 
then 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/d/d033/d033890/d03389046.png" /></td> </tr></table>
+
$$
 +
\lim\limits _ {y \rightarrow y _ {0} }  \lim\limits _ {x \rightarrow x _ {0} } \
 +
f ( x , y )  = \lim\limits _ {y \rightarrow y _ {0} } \
 +
\phi ( y)
 +
$$
  
 
exists and is equal to the double limit of the function.
 
exists and is equal to the double limit of the function.
  
Using the concept of a neighbourhood, the following form may be assigned to the definition of the double limit of a function: Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033890/d03389047.png" /> be a limit point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033890/d03389048.png" /> of a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033890/d03389049.png" /> or the symbol <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033890/d03389050.png" />, the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033890/d03389051.png" /> being unbounded in the latter case, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033890/d03389052.png" /> be a number or one of the symbols <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033890/d03389053.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033890/d03389054.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033890/d03389055.png" />. Then
+
Using the concept of a neighbourhood, the following form may be assigned to the definition of the double limit of a function: Let $  a $
 +
be a limit point $  ( x _ {0} , y _ {0} ) $
 +
of a set $  E $
 +
or the symbol $  \infty $,  
 +
the set $  E $
 +
being unbounded in the latter case, and let $  A $
 +
be a number or one of the symbols $  \infty $,  
 +
$  + \infty $,  
 +
$  - \infty $.  
 +
Then
  
<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/d/d033/d033890/d03389056.png" /></td> </tr></table>
+
$$
 +
= \lim\limits _ {( x , y ) \rightarrow ( x _ {0} , y _ {0} ) } \
 +
f ( x , y )
 +
$$
  
if for any neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033890/d03389057.png" /> of the point or symbol <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033890/d03389058.png" /> there exists a neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033890/d03389059.png" /> of the number or symbol <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033890/d03389060.png" /> such that for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033890/d03389061.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033890/d03389062.png" />, the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033890/d03389063.png" /> is satisfied. In this form the definition of the double limit of a function is applied to the case when the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033890/d03389064.png" /> is defined on the product of two topological spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033890/d03389065.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033890/d03389066.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033890/d03389067.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033890/d03389068.png" />, while the values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033890/d03389069.png" /> also belong to a topological space.
+
if for any neighbourhood $  O _ {A} $
 +
of the point or symbol $  A $
 +
there exists a neighbourhood $  O _ {a} $
 +
of the number or symbol $  a $
 +
such that for all $  ( x , y ) \in E \cap O _ {a} $,
 +
$  ( x , y ) \neq a $,  
 +
the condition $  f ( x , y ) \in O _ {A} $
 +
is satisfied. In this form the definition of the double limit of a function is applied to the case when the function $  f $
 +
is defined on the product of two topological spaces $  X $
 +
and $  Y $
 +
and $  x \in X $,  
 +
$  y \in Y $,  
 +
while the values of $  f ( x , y ) $
 +
also belong to a topological space.

Latest revision as of 19:36, 5 June 2020


The double limit of a sequence, the limit of a double sequence, $ \{ x _ {mn} \} $, $ m , n = 1 , 2 \dots $ is a number $ a $ which is defined as follows: For any $ \epsilon > 0 $ there exists an $ N _ \epsilon $ such that for all $ m , n > N _ \epsilon $ the equality

$$ | x _ {mn} - a | < \epsilon $$

is valid. The notation is

$$ a = \lim\limits _ {m , n \rightarrow \infty } x _ {mn} . $$

If for any $ \epsilon > 0 $ there is an $ N _ \epsilon $ such that for all $ m , n > N _ \epsilon $ the inequality $ | x _ {mn} | > \epsilon $ is fulfilled, then the sequence $ x _ {mn} $ has infinity as its limit:

$$ \lim\limits _ {m , n \rightarrow \infty } x _ {mn} = \infty . $$

The infinite limits

$$ \lim\limits _ {m,n \rightarrow \infty } x _ {mn} = + \infty \ \textrm{ and } \ \ \lim\limits _ {m,n \rightarrow \infty } x _ {mn} = - \infty $$

are defined in the same manner. The double limit of a sequence is a special case of the double limit of a function over a set, namely when this set consists of the points on the plane with integer coordinates $ m $ and $ n $. Accordingly, the double limit of a sequence is connected with its repeated limits as in the general case.

The double limit of a function is the limit of a function of two variables, defined as follows. Let the function $ f ( x , y ) $ be defined on a set $ E $ in the $ X Y $- plane, and let $ ( x _ {0} , y _ {0} ) $ be a limit point of it (cf. Limit point of a set). A number $ A $ is said to be the double limit of the function $ f ( x , y ) $ at the point $ ( x _ {0} , y _ {0} ) $, or as $ ( x , y ) \rightarrow ( x _ {0} , y _ {0} ) $, if for any $ \epsilon > 0 $ there exists a $ \delta > 0 $ such that for all the points $ ( x , y ) \in E $ the coordinates of which satisfy the inequalities

$$ 0 < | x - x _ {0} | < \delta ,\ \ 0 < | y - y _ {0} | < \delta , $$

the inequality

$$ | f ( x , y ) - A | < \epsilon $$

is valid. In such a case the limit is written as

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

The double limit of a function may be formulated in terms of the limit of a sequence:

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

if for any sequence

$$ ( x _ {n} , y _ {n} ) \rightarrow ( x _ {0} , y _ {0} ), $$

$$ ( x _ {0} , y _ {0} ) \neq ( x _ {n} , y _ {n} ) \in E ,\ n = 1, 2 \dots $$

the condition

$$ \lim\limits _ {n \rightarrow \infty } f ( x _ {n} , y _ {n} ) = A $$

is satisfied. The double limit of a function as its arguments tend to infinity, as well as the definitions of infinite double limits of a function, are formulated in a similar manner. There exists a connection between the double limit of a function and the repeated limit of a function at a point $ ( x _ {0} , y _ {0} ) $ or at $ \infty $: Let $ x _ {0} $ and $ y _ {0} $ be limit points (finite or infinite) of the real subsets $ X $ and $ Y $, and let $ E = X \times Y $. If for a function a finite or infinite double limit

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

exists, and if for any $ y \in Y $ there is a finite limit

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

then the repeated limit

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

exists and is equal to the double limit of the function.

Using the concept of a neighbourhood, the following form may be assigned to the definition of the double limit of a function: Let $ a $ be a limit point $ ( x _ {0} , y _ {0} ) $ of a set $ E $ or the symbol $ \infty $, the set $ E $ being unbounded in the latter case, and let $ A $ be a number or one of the symbols $ \infty $, $ + \infty $, $ - \infty $. Then

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

if for any neighbourhood $ O _ {A} $ of the point or symbol $ A $ there exists a neighbourhood $ O _ {a} $ of the number or symbol $ a $ such that for all $ ( x , y ) \in E \cap O _ {a} $, $ ( x , y ) \neq a $, the condition $ f ( x , y ) \in O _ {A} $ is satisfied. In this form the definition of the double limit of a function is applied to the case when the function $ f $ is defined on the product of two topological spaces $ X $ and $ Y $ and $ x \in X $, $ y \in Y $, while the values of $ f ( x , y ) $ also belong to a topological space.

How to Cite This Entry:
Double limit. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Double_limit&oldid=46770
This article was adapted from an original article by L.D. Kudryavtsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article