Examples
Examples abound, one of the simplest being that for a double sequence
- am,n
it is not necessarily the case that the operations of taking the limits as m → ∞ and as n → ∞ can be freely interchanged. For example take
- am,n = 2m − n
in which taking the limit first with respect to n gives 0, and with respect to m gives ∞. Replacing all values greater than 1 by 1, the limits can be made to be 0 or 1 according to the order.
Many of the fundamental results of infinitesimal calculus also fall into this category: the symmetry of partial derivatives, differentiation under the integral sign, and Fubini's theorem deal with the interchange of differentiation and integration operators. One of the major reasons why the Lebesgue integral is used is that theorems exist, such as the dominated convergence theorem, that give perspicuous reasons why integration and infinite summation can be interchanged.
Read more about this topic: Interchange Of Limiting Operations
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