Which infinity is meant in limits? For example, when we write $\lim_{x\rightarrow \infty} f(x)$ - which infinity is meant and why? Countable? If uncountable - which and why?
 A: There is no need to ask about countable or uncountable infinity. The symbol $\infty$ here is used with the following precise meaning.

Let $f$ be defined on some ray $(a,\infty)$. We say that $$\lim_{x\to\infty}f(x)=A$$ if there exists $A\in\Bbb R$ such that for each $\epsilon >0$ there exists an $M>0$ for which $$x>M\implies |f(x)-A|<\epsilon$$

That is, for any $\epsilon >0$ we're given, we can take $x$ large enough to make $f$ as close as we wish to $A$.
The more "dramatic" (symbolically) $$\lim_{x\to\infty}f(x)=\infty$$ means precisely that

... for each $N>0$ there exists $M>0$ such that $x>M\implies f(x)>N$.

That is, we can make $f(x)$ as large as we want by taking $x$ large enough.
ADD Compare the above to the notation $$(a,\infty)$$ I used in it. We're not wondering about any "infinity" but just about the set of numbers $x>a$. The symbol $\infty$ is convenient and intuitive. Brian uses $(a,\to)$ instead!
A: The $\infty$ in that limit does not refer to an infinite cardinality at all. The expression
$$\lim_{x\to\infty}f(x)=L$$
is simply an abbreviation for the following statement:

$\qquad\qquad$for each $\epsilon>0$ there is an $x_\epsilon\in\Bbb R$ such that $|f(x)-L|<\epsilon$ whenever $x\ge x_\epsilon$.

As you can see, there is no infinite anywhere in that statement. The ‘$x\to\infty$’ in the limit notation is a reminder that we’re talking about what happens when $x$ is very large; it does not refer to a specific entity $\infty$.
(There is in fact a way to interpret this $\infty$ as a specific entity: one can replace $\Bbb R$ with the so-called extended reals, which include two new points $\infty$ and $-\infty$. In effect this adds an endpoint at each end of the real line. But these objects, despite their standard names $\pm\infty$, are simply points in an extended space, not cardinal numbers that can be used for counting infinite collections.)
A: None of the above. That's why it neither says $\omega$,nor $\aleph_0$, nor $\mathbf c$.
Rather, $\infty$ should be considered as a symbol.
There's no infinity really used in the definition:
$$\lim_{x\to\infty}f(x)=c\iff\forall\epsilon>0\colon\exists M\in\mathbb R\colon\forall x>M\colon |f(x)-c|<\epsilon.$$
A: The infinity here is a point at infinity in the two-point compactification $\mathbb{R} \cup \{ -\infty, +\infty \}$ of the real line. This is just one of many meanings of "infinity" in mathematics; see, for example, this math.SE question. 
A: The infinity in the definition of limit has nothing to do with cardinality. Cardinalities are related to sizes of sets. Limits of functions have nothing to do with sizes, but rather with behaviour as $x$ approaches something. So, the relevant notion of infinity lurking under the surface is that of a directed set, and in the case of limits in metric spaces the simpler notion of a sequence. A directed set is a poset $(A,\le)$ such $A\ne \emptyset$ and for all $a,b\in A$ there exists $c\in A$ with $a\le c$ and $b\le c$. A particular type of ordered set is the set $\omega$ of the natural numbers, with their usual ordering. 
As you probably know, in the context of metric spaces, a function $f:X\to Y$ satisfies $\lim _{x\to x_0}f(x)=L$ if, and only if, for all sequences $x:\omega \to X$ with $\lim _{n\in \omega}x=x_0$ holds that $\lim _{n\in \omega}f(x)=L$. It is the order type of $\omega$ that gives rise to the usual meaning of the expressions $\lim _{n\in \omega }x=x_0$ and $\lim _{n\in \omega }f(x)=L$. 
In situations other than metric spaces to correctly capture limits of functions it is required to consider 'sequences' over directed sets other than $\omega$. These are called nets. The reason that in metric spaces sequences suffice is that the metric takes values in $[0,\infty ]$ and limit behaviour is sensitive to 'getting close to $0$'. The set $\{\frac{1}{n}\mid n\in \mathbb N\}$ gives rise to a discrete directed set (after inverting the order) which is why sequences are all one needs to properly investigate 'getting close to $0$'. 
So, the important underlying infinity when discussing limits is not at all related to cardinalities, but rather how a quantity is allowed to approach something - i.e., nets. Nets require directed sets, and the $\infty $ in the usual definitions of limits refers to the (infinite) directed set $\mathbb N$. In short, it is not cardinals but rather ordinals that are at play.
