I need some help to clarify with the idea of pointwise convergence.
The condition for pointwise convergence is usually given that if the limit of $f_n$ exists for each $x$ $\in$ $A$.
In Stephen Abbott's Understanding Analysis book, to show comparison with uniform convergence, the pointwise convergence definition is given as: " $f_n$ converges pointwise on $A$ to a limit function $f$ defined on $A$ if, for every $\epsilon>0$ and $x$ $\in$ $A$, there exists an $N$ $\in$ $\mathbb{N}$ (can be dependent on $x$) such that $|f_n(x)-f(x)|<\epsilon$ whenever $n$ $\geq$ $N$."
Taking an example of $x/(x+n)$, $x\geq0$, we can see the limit is $0$ holds for all $x\geq0$, but when evaluating it in terms of finding epsilon, $|f_n(x)-f(x)|$ can only be bounded with $\epsilon\geq1$, isn't this contradict with the book's definition? but how can we explain the condition of limit exists? Does this example have pointwise convergence after all?
The difference between pointwise and uniform convergence is that the choice of $N \in \mathbb N$ depends on the choices of both $x$ and $\varepsilon$ in the case of pointwise convergence, while it only depends on $\varepsilon$ in the case of uniform convergence. That is, in the case of uniform convergence, there will exist an $N$ for each $\varepsilon$ such that convergence is “uniform” (or $N$ is the same) for each $x$.
In your example, only pointwise convergence is possible, since the choice of $N$ will always depend on both $x$ and $\varepsilon$.