What is the general limit theorem? There are simple limit theorems like http://archives.math.utk.edu/visual.calculus/1/limits.18/
But they are just special cases. I am quite sure there is an established general result for them.
In other words,
for what conditions of h does 
$$\lim_{x\rightarrow a}h(f(x),g(x))=h(\lim_{x\rightarrow a}f(x),\lim_{x\rightarrow a}g(x))$$
hold?
For what conditions of h does
$$\lim_{x\rightarrow a}h(f(x))=h(\lim_{x\rightarrow a}f(x))$$
hold?
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mixedmath's answer: we have
$$\lim_{x\rightarrow a}h(f(x),g(x))=h(\lim_{x\rightarrow a}f(x),\lim_{x\rightarrow a}g(x))$$
if $h$ is continuous,
and we also have 
$$\lim_{x\rightarrow a}h(f(x))=h(\lim_{x\rightarrow a}f(x))$$
if $h$ is continuous.
 A: This is a definition of continuity. See for example Wikipedia's sequence definition of continuity.
The second case is easier to talk about.
Supposing that $f(x) \to f(a)$, then you can think of any sequence of terms $x_n \to a$ and think of $f_n:= f(x_n)$ as a sequence of terms that goes to $f_a = f(a)$, where I'm using subscripts to emphasize that I'm thinking of these as just numbers and not results of a function. Then the statement that $\lim_{n \to a} h(f_n) = h(f_a)$ for any sequence $f_n \to f_a$ is exactly the statement that $h$ is continuous at $f_a$.
Similarly for your first question.
So you are correct to think that everything in your linked page falls under a larger umbrella. For example, the fact that the addition function $p(x,y) = x+y$ is continuous (which is easy to show, and a reasonable and approachable exercise if not immediately obvious) gives us that $\lim_{x \to a} p(f(x), g(x)) = p(\lim f(x), \lim g(x)$. Similarly for subtraction, multiplication, etc. 
And now you might ask - is there a general result on why functions like addition, subtraction, and multiplication are continuous. You might realize that each of these are polynomials of the inputs, and all polynomials are continuous [although this is typically shown by first knowing that the sum of continuous functions is continuous, which is circling around being circular]. You link also uses that exponentiation and taking roots are continuous functions, which don't umbrella as nicely.
