For example I have this:
$\lim\limits_{x \to x_o} f(x)$
and also
$\lim\limits_{u \to 0} f(x_0 + u)$
But i'm not really sure how to treat this second expression, is it a different function? what does it mean to add terms inside the parenthesis and take the limit?
That is the condensed form of the following : define the function $g(u) = f(x_0+u)$. So, to compute $g(u)$ for any $u$, we first compute $x_0+u$ and then apply $f$ on the output.
Then, the quantity $\lim_{u \to 0} f(x_0+u)$ means the same as $\lim_{u \to 0} g(u)$. So it is the limit of a different function $g$, but rather than write the definition of $g$ separately and use it, we just condense it in one line as $\lim_{u \to 0} f(x_0+u)$.
The "addition inside the parentheses" is part of the definition of $g$, as was explained earlier.
As you go along, you will find more of these expressions, where you will have $$\lim_{some\ variable \to some\ value} \{\mathrm{something\ depending\ on \ f(x) \ and \ other\ things}\}$$
these all mean that the "something depending on $f(x)$" is a function in the variable under the limit, and the quantity in question is the limit of this function as the variable approaches the given value.