limit function
When a sequence of functions converges, the destination is itself a function — the limit function. Instead of a single number serving as the limit of a sequence of numbers, here a whole function f serves as the limit of a sequence of functions f_n. It is built one value at a time: at each input x you take the ordinary numerical limit of f_n(x).
Definition: if f_n -> f pointwise on E, the limit function is f(x) = lim_{n->infinity} f_n(x), defined value by value for each x in E. For a series, the limit function is the sum s(x) = lim_{N->infinity} sum_{n=1}^N f_n(x), i.e. the limit of the partial sums. The same f is reached whether the convergence is pointwise or uniform; the MODE of convergence changes nothing about WHICH function f is, only how well f inherits the good properties of the f_n.
The whole subject is a study of what the limit function inherits. Under mere pointwise convergence f can be a wild object — discontinuous, non-integrable, or with the 'wrong' integral and derivative. Under uniform convergence f inherits continuity and (with care) integrability and differentiability. So when you meet a limit function, the first question is always: in what sense did the f_n converge to it?
Let f_n(x) = nx/(1 + n^2 x^2) on [0, 1]. For each fixed x > 0, f_n(x) -> 0; and f_n(0) = 0. So the limit function is f(x) = 0 identically. Yet f_n(1/n) = 1/2 for all n, so the graphs always have a bump of height 1/2 that slides left toward 0 — convergence to the zero limit function is pointwise but not uniform.
A sliding bump: pointwise limit 0, but the bump never shrinks.