Final answer:
The Sequential Continuity Theorem allows for the application of a continuous function f(x) to the limit of a converging sequence An, resulting in lim f(An) = f(lim An) = f(L) where L is the limit of the sequence.
Step-by-step explanation:
The theorem you are referring to is often known as the Sequential Continuity Theorem. To use this theorem, we first need a sequence (An) that converges to a limit L.
Then, we consider a function f(x) that is continuous at L. According to the theorem, if these conditions are met, we can say that the limit of the function of the sequence as n approaches infinity is equal to the function of the limit of the sequence, that is:
lim (as n → ∞) f(An) = f(lim (as n → ∞) An) = f(L)
This theorem allows us to apply continuous functions to limits of sequences, significantly simplifying the analysis of the behavior of complex sequences, especially when dealing with limits approaching infinity or when summing infinite series.