Why is the interpretation of infinite union of sets as a limit so incorrect (or even dangerous)? In some reputable notes on a Probability course at the IIT I read:

When the index set $\mathcal I$ is a finite set, say $\mathcal I = {1, 2, 3}$ the definition of union given above coincides with the “middle-school” understanding of unions, i.e., taking the union of sets one-at-a-time. For example, $\bigcup_{i=1}^3 A_i= A_1 \bigcup A_2 \bigcup A_3.$ However, this “one-by-one” interpretation completely breaks down when the index set $\mathcal I$ is infinite. For example when $\mathcal I = \mathbb N,$ the union $\bigcup_{i=1}^\infty A_i$ does not have any interpretation in terms of taking unions one by one, till infinity. After all, there is no $A_∞$ in the family $\{A_i, i ∈ \mathbb N\},$ and there is no notion of “limiting unions”. Thus, $\bigcup_{i=1}^\infty A_i$ should be interpreted just as Definition 1.3 says: it is the set of all elements contained in at least one of the $A_i, i ∈ \mathbb N.$ In order to avoid the (dangerous) temptation to interpret $\bigcup_{i=1}^\infty A_i$ as some sort of a limit of finite, “one-by-one” unions, a better notation would be to use $\bigcup_{i\in \mathbb N} A_i,$ instead of the potentially misleading but more commonly used notation $\bigcup_{i=1}^\infty A_i.$

What sort of misunderstanding would the approach of this union as a limit result in?
 A: You're really looking for $\limsup$ and $\liminf$ of sets. Let $(A_n)_{n\geq 0}$ be a countable family of sets. Then $$\limsup_{n\to \infty}A_n = \bigcap_{n\geq 0} \bigcup_{k \geq n} A_k \qquad \mbox{and}\qquad \liminf_{n\to \infty} A_n = \bigcup_{n\geq 0} \bigcap_{k \geq n}A_k.$$If these two sets coincide, then we can write $\lim_{n\to \infty} A_n$.
Example: if $A_n=[-n,n]$ for all $n$, then $\lim_{n\to \infty}[-n,n]=\Bbb R$, as you would expect.
In general, if $(A_n)_{n\geq 0}$ in non-increasing or non-decreasing, the limit exists and it equals their union or intersection.
The bottom line is that in the same way that the limit of a numerical real sequence exists if and only if both the limsup and liminf exist and are equal, this is what you have to be careful with when dealing with sets.
A: The reason viewing infinite unions as limits is technically incorrect in this context is for the exact reason your quoted passage says: there is no definition of a "limit" when it comes to taking the union of sets in this context. When you think of, for example
$$
\lim_{x\to a}f(x)
$$
this has a definition. ${\forall\ \epsilon>0\ \exists\ \delta>0}$... yadda yadda yadda.
But when it comes to sets, we don't have any such notion. Nor do we need it. You can simply say that ${x \in \bigcup_{i \in \mathbb{N}}A_i}$ if and only if ${x \in A_i}$ for some ${i \in \mathbb{N}}$. That's it. That's all we need to define the infinite union.
EDIT: @Peter Morfe has corrected me in the comments of this post. There is such a notion defined, and it is useful in a lot of contexts. The main point still stands, though, that we do not need a notion of set limits to define at least a basic notion of the union and intersection of sets. In fact, for the union/intersection of uncountably infinitely many sets, the standard definition I referenced is the one you need to use.
