Difference between revisions of "Double limit"
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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 | ||
− | + | $$ | |
+ | a = \lim\limits _ {m , n \rightarrow \infty } x _ {mn} . | ||
+ | $$ | ||
− | If for any | + | 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 | 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 | + | 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 | + | 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 | ||
− | < | + | $$ |
+ | 0 < | x - x _ {0} | < \delta ,\ \ | ||
+ | 0 < | y - y _ {0} | < \delta , | ||
+ | $$ | ||
the inequality | the inequality | ||
− | + | $$ | |
+ | | f ( x , y ) - A | < \epsilon | ||
+ | $$ | ||
is valid. In such a case the limit is written as | is valid. In such a case the limit is written as | ||
− | + | $$ | |
+ | \lim\limits _ {( x , y ) \rightarrow ( x _ {0} , y _ {0} ) } \ | ||
+ | f ( x , y ) = A . | ||
+ | $$ | ||
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: | ||
− | + | $$ | |
+ | A = \lim\limits _ {( x, y ) \rightarrow ( x _ {0} , y _ {0} ) } \ | ||
+ | f ( x , y ) , | ||
+ | $$ | ||
if for any sequence | 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 | 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|repeated limit]] of a function at a point | + | 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 | ||
− | + | $$ | |
+ | \lim\limits _ {( x , y ) \rightarrow ( x _ {0} , y _ {0} ) } \ | ||
+ | f ( x , y ) | ||
+ | $$ | ||
− | exists, and if for any | + | 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 | 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. | 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 | + | 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 | + | 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.
Double limit. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Double_limit&oldid=46770