Question

Show tuat the double limit of a function may not exist even if the repeated limit exist and are equal

Answer 1

Answer:

is below.

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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  of two variables  and  be defined on a set of the form , , , and let  and  be limit points of the sets  and , respectively, or the symbol  (if  or ,  and, respectively,  may be infinities with signs: , ). If for any fixed  the limit

(1)

exists, and for  the limit



exists, then this limit is called the repeated limit

(2)

of the function  at the point . Similarly one defines the repeated limit

(3)

If the (finite or infinite) double limit

(4)

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

If for each  the finite limit (1) exists, for each  the finite limit



exists, and for  the function  tends to a limit function  uniformly in , then both the repeated limits (2) and (3) exist and are equal to one another.

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

and the repeated limits

are called the repeated limits of the double sequence.

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