Repeated limit

A limit of a function of several variables in which the passage to the limit is performed successively in the different variables. Let, for example, a function $ f $ of two variables $ x $ and $ y $ be defined on a set of the form $ X \times Y $, $ x \in X \subset \mathbf R ^ {m} $, $ y \in Y \subset \mathbf R ^ {n} $, and let $ x _ {0} $ and $ y _ {0} $ be limit points of the sets $ X $ and $ Y $, respectively, or the symbol $ \infty $( if $ m = 1 $ or $ n = 1 $, $ x _ {0} $ and, respectively, $ y _ {0} $ may be infinities with signs: $ + \infty $, $ - \infty $). If for any fixed $ y \in Y $ the limit

$$ \tag{1 } \phi ( y) = \lim\limits _ {x \rightarrow x _ {0} } f( x, y) $$

exists, and for $ \phi ( y) $ the limit

$$ \lim\limits _ {y \rightarrow y _ {0} } \phi ( y) $$

exists, then this limit is called the repeated limit

$$ \tag{2 } \lim\limits _ {y \rightarrow y _ {0} } \lim\limits _ {x \rightarrow x _ {0} } f( x, y) $$

of the function $ f $ at the point $ ( x _ {0} , y _ {0} ) $. Similarly one defines the repeated limit

$$ \tag{3 } \lim\limits _ {x \rightarrow x _ {0} } \lim\limits _ {y \rightarrow y _ {0} } f( x, y). $$

If the (finite or infinite) double limit

$$ \tag{4 } \lim\limits _ {( x, y) \rightarrow ( x _ {0} , y _ {0} ) } f( x, y) $$

exists, and if for any fixed $ y \in Y $ the finite limit (1) exists, then the repeated limit (2) also exists, and it is equal to the double limit (4).

If for each $ y \in Y $ the finite limit (1) exists, for each $ x \in X $ the finite limit

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

exists, and for $ x \rightarrow x _ {0} $ the function $ f( x, y) $ tends to a limit function $ \phi ( y) $ uniformly in $ y \in Y $, then both the repeated limits (2) and (3) exist and are equal to one another.

If the sets $ X $ and $ Y $ are sets of integers, then the function $ f $ is called a double sequence, and the values of the argument are written as subscripts:

$$ f( m, n) = u _ {mn} , $$

and the repeated limits

$$ \lim\limits _ {n \rightarrow \infty } \lim\limits _ {m \rightarrow \infty } u _ {mn} \ \textrm{ and } \ \lim\limits _ {m \rightarrow \infty } \lim\limits _ {n \rightarrow \infty } u _ {mn} $$

are called the repeated limits of the double sequence.

The concept of a repeated limit has been generalized to the case where $ X $ and $ Y $ and the set of values of the function $ f $ are subsets of certain topological spaces.