What is a general definition of a limit?

[Lotto]

TL;DR

I was given this definiton of a limit:

Let us have a function $f: \mathbb{R} \rightarrow \mathbb{C}$. Let $x_0 \in \mathbb{R}^*$ and $L \in \mathbb{R}^*$. We say $L$ is a limit of a function $f$ for $x$ goes to $x_0$ if

$\forall \varepsilon >0 \, \exists \delta >0 \, \forall x \in P_{\delta}(x_0): f(x) \in U_{\varepsilon}(L)$

($P$ and $U$ are neighborhoods)

Is this definition valid for every type of limit?

I suppose that it is because we are in extended real numbers. But the definition of a limit when $x_0 = \infty$ and let's say $L=\infty$ is different. Why are these definitions equivalent? Isn't the key that $U_{\varepsilon}(\infty)=\left(\frac {1}{\varepsilon},\infty\right)$?

 

[fresh_42]

What does "I was given" mean?

That definition defines a continuous function $f,$ not a limit. What both connects them is
$$
\lim_{n \to \infty} f(x_n)=f(\lim_{n \to \infty}x_n)=f(x_0)
$$
so the limits both exist, i.e. are finite, and if $\mathbf f$ is continuous. This can be used as an equivalent definition of a at $x_0$ continuous function, but you need the definition of a limit first.

It doesn't make much sense for infinities since they have no neighborhoods. That's why converges to a finite number and grows beyond all finite numbers are treated differently.

A limit $x_0$ of a sequence $(x_n)_{n\in\mathbb{N}}$ is:
$$
\forall \, \varepsilon> 0\,\exists \,N(\varepsilon)\in \mathbb{N}\, \forall\, n>N(\varepsilon)\, : \, x_n\in U_\varepsilon(x_0).
$$
and in case $x_0=\infty$
$$
\forall\,M\in\mathbb{R}\,\exists\, N(M)\in \mathbb{N} \, \forall \, n>N(M)\,: \, x_n >M.
$$
and likewise for $x_0=-\infty .$