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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|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.