5 Differentiation of Sequences and Series of Functions

The case x < a can be handled similarly, and so limx→a f (x)g(x) = L. #

The following result is often of great practical use.

Theorem 1.25 For n ∈ N, consider the sequence of functions fn : (a, b)→ R such that the following hold:

(i) For some x0 ∈ (a, b), the limit lim fn(x0) exists.