What does "the sequence of functions has limit in R" mean?

[cbarker1]

TL;DR Summary

I am trying understand a condition in a set notion.

Suppose f1,f2... is a sequence of functions from a set X to R. This is the set T={x in X: f1(x),... has a limit in R}. I am confused about what is the meaning of the condition in the set. Is the limit a function or a number value? Why?

 

[FactChecker]

I would have to say both. The condition "f1(x),... has a limit in R" is applied for one value of $x \in R$ at a time, so it gives a single value. But it is applied for every $x \in X$, so that gives a function on $X$.