Double limit

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.