Double limit

The double limit of a sequence, the limit of a double sequence, , is a number which is defined as follows: For any there exists an such that for all the equality

is valid. The notation is

If for any there is an such that for all the inequality is fulfilled, then the sequence has infinity as its limit:

The infinite limits

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 and . 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 be defined on a set in the -plane, and let be a limit point of it (cf. Limit point of a set). A number is said to be the double limit of the function at the point , or as , if for any there exists a such that for all the points the coordinates of which satisfy the inequalities

the inequality

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

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

if for any sequence

the condition

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 or at : Let and be limit points (finite or infinite) of the real subsets and , and let . If for a function a finite or infinite double limit

exists, and if for any there is a finite limit

then the repeated limit

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 be a limit point of a set or the symbol , the set being unbounded in the latter case, and let be a number or one of the symbols , , . Then

if for any neighbourhood of the point or symbol there exists a neighbourhood of the number or symbol such that for all , , the condition is satisfied. In this form the definition of the double limit of a function is applied to the case when the function is defined on the product of two topological spaces and and , , while the values of also belong to a topological space.