Equivalence of Rudin's Definition 4.33 (infinite limits and limits at infinity)?

Question

In Rudin's "Principles of Mathematical Analysis", Definition 4.33 (p98) states:

Definition 4.33 $\quad$ Let $f$ be a real function defined on $E.$ We say that $f(t)\to A$ as $t\to x$, where $A$ and $x$ are in the extended real number system, if for every neighborhood $U$ of $A$ there is a neighborhood $V$ of $x$ such that $V \cap E$ is not empty, and such that $f(t)\in U$ for all $t\in V\cap E, t\ne x.$

I'm a little confused however by the remark right beneath this definition:

A moment's consideration will show that this coincides with Definition 4.1 when $A$ and $x$ are real.

My question is actually three-fold:

1) In Definition 4.1 (quoted below), it is explicitly stated that $x$ ($p$) has to be a limit point of $E$. Can this be inferred from Definition 4.33? It appears to me that according to Definition 4.33 the limit of a function can also be defined even if $x$ is an isolated point of $E.$ For example, let $E=\{0\}\cup \{1\} \cup [3,4]$, and $f(0)=f(1)=0, f(t)=t, t\in [3,4]$. Then with $V$ being a neighborhood of radius $2$ around $0$, we have $f(t)\to 0$, as $t\to 0,$ don't we?

2) Similarly, for limits at infinity, i.e. $x=\infty$, Definition 4.33 doesn't require the domain $E$ of $f$ to be unbounded, does it? For example, if $f(x)=1$ for $x\in[0,1]$, then we may also say $f(x)\to 1$ as $x\to \infty$? (e.g. with $V=(0.5, \infty)$ we clearly satisfy Definition 4.33.)

3) Should we or should we not add these assumptions (or implicitly assume them) when using Definition 4.33? (i.e. $x$ has to be limit point of $E$, or $E$ is unbounded above or below.)

Thanks a lot!

Definition 4.1 $\quad$ Let $X$ and $Y$ be metric space; suppose $E \subset X$, $f$ maps $E$ into $Y$, and $p$ is a limit point of $E$. We write $f(x)\to q$ as $x\to p$, or $\lim_{x\to p}f(x)=q$ if there is a point $q \in Y$ with the following property: For every $\epsilon > 0$ there exists a $\delta > 0$ such that $d_Y(f(x),q)<\epsilon$ for all points $x\in E$ for which $0