The limit of the sum is the sum of the limits I was wondering why the statement in the title is true only if the functions we are dealing with are continuous.
Here's the context (perhaps not required): 

(The upper equation there is just a limit of two sums, and the lower expression is two limits of those two sums.), and if anyone wonders, that's the original source (a pdf explaining the proof of the product rule).
P.S. In the context it's given that $g$ and $f$ are differentiable, anyway I only provided it to illustrate the question; my actual question is simply general.

The limit of the sum is not always equal to the sum of the limits, even when the individual limits exist.
For example:
Define $h(i)=1\sqrt{(n^2)+i}$. 
For each $i=1,\cdots,n$, the limit of $h(i)$ is zero as n goes to infinity.
But the limit of the sum $[h(1)+h(2)+\cdots+h(n)]$ as n goes infinity is not zero.
The limit of this sum is actually $1$, using the sandwich theorem.
