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.