Consider this function:
$$f : \{0\}\cup [1,100) \to \Bbb R : x \mapsto \begin{cases}0& x=0\\1&x\ne 0\end{cases}$$
Suppose we are looking for $\lim_\limits{x\to 0} f(x)$. The "$U$" in this case is $U = \{0\}\cup [1,100)$. It turns out that $$\lim_\limits{x\to 0} f(x) = 1$$
Because for any $\epsilon > 0$, we can set $\delta = 50$. then $B(0,50)\cap U(x \ne 0) = [1,50)$ and for all $x \in [1,50), f(x) = 1 \in B(1,\epsilon)$.
But it gets even better. It is also true that
$$\lim_\limits{x\to 0} f(x) = -1001$$
because for any $\epsilon > 0$, we can set $\delta = \frac 12$. Then $B(0,1/2)\cap U(x \ne 0) = \emptyset$, and since $x \in \emptyset$ is always false, it is certainly true that $x\in \emptyset \implies f(x) \in B(-1001,\epsilon)$. Of course, I could have used any number as the limit and followed the same argument.
This demonstrates why we don't define limits at isolated points. In the above example, the $\lim_\limits{x\to 1} f(x) = 1$ and nothing else, because $1$ is not isolated. But does $U$ actually have to be open? No. The $U$ in the example is perfectly fine for defining limits of $f$ for any point in $U$ except $0$. It is even fine for defining $\lim_\limits{x\to 100} f(x)$ even though $100 \notin U$, which is exactly the same thing as we do when defining limits at $\infty$, which nobody complains about. We can define the limit of a function at any point on the boundary of its domain in some known larger space (in this case $\Bbb R$).
Now let me digress for a moment because practically every time I've made this point, someone eventually comes along proclaiming "That's wrong, because you don't know what $f$ is doing on the other side of $1$". But they are mistaken. We do know exactly what $f$ is doing on the other side of $1$. It is doing nothing. $U$ is the domain of $f$. Outside of $U, f$ is not defined. You may have some extension of $f$. But that is not $f$. The domain of a function is part of its definition. And it is $f$ whose limit we are defining, not some extension.
So why is $U$ restricted to open sets? Not because the definition of the limit required it. No, the author resticted to open domains simply because that is all they needed for what they are planning to do. Their goal is something other than defining limits in maximum generality (doing so goes far deeper than this definition). They are driving for a certain set of tools and ideas, and for those tools, defining the limit in an open domain is sufficient. To spend time discussing what happens at boundary points would just be an unnecessary distraction from their purpose.
