Can I multiply *into* limits? I learned a lot yesterday and it requires me to overhaul the way I think about derivatives.  So I have something worked out but it relies on the answer to this question.
EDIT: Basically I need a simple proof that says $b \cdot [\lim_{h \to a} g(h)] = \lim_{h \to a} [b\cdot g(h)]$
 A: In the expression
$$\color{red}{h} \lim_{\color{green}{h} \to 0} \frac{a}{\color{green}{h}}$$
$\color{red}{h}$ and $\color{green}{h}$ are different variables. You could bring $\color{red}{h}$ inside the limit, but if you did, you can't cancel it with $\color{green}{h}$. This limit is either 0 (if $a=0$) or it doesn't exist.
In the expression
$$\color{red}{h} \lim_{\color{green}{h} \to 0} \frac{a}{\color{red}{h}}$$
you could bring $\color{red}{h}$ into the limit, but there isn't really any problem, because you already know
$$\lim_{\color{green}{h} \to 0} \frac{a}{\color{red}{h}} = \frac{a}{\color{red}{h}}$$
This expression, however, is nonsense:
$$\color{green}{h} \lim_{\color{green}{h} \to 0} \frac{a}{\color{green}{h}}$$
The variable $\color{green}{h}$ introduced in the limit expression only "exists" inside the limit expression. It's simply not allowed to appear outside of the limit expression on the left like that.
Of course, you usually don't have the colors to help distinguish variables.* If a variable $h$ already has meaning in your work, it is very confusing to introduce a new variable called $h$, e.g. by writing a limit $\lim_{h \to 0} f(h)$, because it's easy to forget which $h$ is which. Use a different letter instead -- e.g. the synonym $\lim_{k \to 0} f(k)$ -- or do something else to distinguish the two versions, such as color, case, font, decorations, et cetera. -- e.g. $\lim_{\mathbf{\hat{h}} \to 0} f(\mathbf{\hat{h}})$.
A: 
Any algebraic manipulation going on inside the brackets of the limit is fine

Mathematically, the content of this statement is that if $f(h) = g(h)$ then $\lim_{h \to a} f(h) = \lim_{h \to a} g(h)$.

can I multiply what is inside the brackets with something outside the brackets

It is a theorem that $k \lim_{h \to a} f(h) = \lim_{h \to a} [k f(h)]$. In fact, this is a simple consequence of a more general theorem:
$$[\lim_{h \to a} f(h)] \cdot [\lim_{h \to a} g(h)] = \lim_{h \to a} [f(h)g(h)].$$
See here for a proof of a very closely related theorem.. whose proof can be very easily adapted to prove this.
A: If $A$ is equal to $B$ then $B$ is equal to $A$.
It folows that if
$$
\lim_{h\to a} (b g(h)) = b \lim_{h\to a} g(h)
$$
then
$$
b\lim_{h\to a} g(h) = \lim_{h\to a} (b g(h)).
$$
