Is there a theorem on the composition of limits? What I mean is, is there some equation relating the limits of two functions with the limit of the composition of a function? And what about 'nested' limits? I know the following theorems about limits:

$\displaystyle \lim_{x \to a} [f(x) + g(x)] = \lim_{x \to a} f(x) + \lim_{x \to a}g(x)$

and

$\displaystyle \lim_{x \to a} [f(x)g(x)] = [\lim_{x \to a} f(x)] [\lim_{x \to a}g(x)]$

and

IF $\displaystyle \lim_{x \to a} f(x) \neq 0$ then

$\displaystyle \lim_{x \to a} \frac{1}{f(x)} = \frac{1}{\lim_{x \to a} f(x)}$

But what about something that would give a simplification of the following limits:

$\displaystyle \lim_{x \to a} f(g(x)) \;\;\; \mathrm{or} \;\;\; \lim_{x \to a} g(f(x)) $

(Assuming that we know $\displaystyle \lim_{x \to a} f(x)$ and $\displaystyle \lim_{x \to a} g(x)$, and assuming these limits exist)

Is there anything we can say about the limit of the composition of these functions? And what about nested limits? Is there anything we can say about an expression like:

$\displaystyle \lim_{x \to a} [ \lim_{x \to a} f(x)]$

? Is there any "rule" about nested limits that can be used to help in the manipulation of such an expression?

Thanks in advance