calculus - When can one apply limit properties?

I'm starting to think you can always apply them, but you might get indeterminate form 0/0. However, I haven't seen this stated in relation to limits, only in some complex derivatives. It's not stated when limit properties are applied.

I had expected that you could apply properties and rules blindly (with any preconditions included as part of the property/rule), but maybe you must also look at the outcome you get in order to know what rules you're allowed to apply? Perhaps it's like division in algebra, where a manipulation might be valid (at "compile time"), but if specific values of variables make a divisor equal zero, then it's undefined (at "runtime")?

I'll give a warmup example, then the one I'm actually concerned about.

1) First example: applying the quotient limit property to $x/x$ as $x\to0$, which should be $1$:

$$
\lim_{x \to 0}\frac{x}{x} = \frac{\lim_{x \to 0}x}{\lim_{x \to 0}x} = \frac{0}{0}
$$

2) Second example: in part of a proof of the product rule (used by Khan at 7:30, and Spivak in Ch.10, Theorem 4, p.45):

$$
\lim_{h\to0}
\frac

{ f(x+h) [ g(x+h) - g(x) ] }
{h}
=
\lim_{h\to0} f(x+h)
\lim_{h\to0}
\frac
{ g(x+h) - g(x) }
{h}
$$

Then using $\lim_{h\to0} f(x+h) = f(x)$:

$$
=
f(x)
\lim_{h\to0}
\frac
{ g(x+h) - g(x) }
{h}
$$

My difficulty is you could apply the same approach to the other factor: $\lim_{h\to0} g(x+h) = g(x)$.

$$
\lim_{h\to0}
\frac
{f(x+h)}
{h}
\lim_{h\to0}
[ g(x+h) - g(x) ]

= \\
\lim_{h\to0}
\frac
{f(x+h)}
{h}
[
\lim_{h\to0}
g(x+h)
-
\lim_{h\to0}

g(x)
]
= \\
\lim_{h\to0}
\frac
{f(x+h)}
{h}
[
g(x)
-

g(x)
]
= \\
\lim_{h\to0}
\frac
{f(x+h)}
{h}
0
=
0

$$

---

Am I wrong to think you should be allowed to apply property limits whenever you like, without condition? (And if there are pre-conditions, they should be part of the property?) Or, is there some aspect relating to $0/0$ (indeterminate form) whose connection to property limits I've somehow missed or not appreciated?

It seems derivatives are about finding a ratio $a/b$, even as $a,b\to0$. But I think limits are meant to be standalone, independent of derivatives.

PS. if my confusion is too fundamental to address in an answer, could you suggest a reference that definitely does address it, please? (It's a lot of work to go through a reference, only to find it doesn't have the answer I need.) Many thanks!