You don't seem to get what I'm saying. Basically, it's not an exception to the theorem because it does not address infinite sums. That's assuming we're talking about the same theorem.

Basically every single convergent series will do the same as this problem does. That's because the nth term has to approach 0 as n increases or the series will diverge.

So in summary, the theorem that says that the sum of limits is the same as the limit of the sums does not hold for an infinite number of functions.