# 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.

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