In mathematics, the study of interchange of limiting operations is one of the major concerns of mathematical analysis. in that two given limiting operations, say L and M, cannot be assumed to give the same result when applied in either order.An algebraist would say that the operations do not commute. The approach taken in analysis is somewhat different. Conclusions that assume limiting operations do 'commute' are called formal. The analyst tries to delineate conditions under which such conclusions are valid; in other words mathematical rigour is established by the specification of some set of sufficient conditions for the formal analysis to hold good. It is relatively rare for such sufficient conditions to be also necessary, so that a sharper piece of analysis may extend the domain of validity of formal results.

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