Here is a (non-exhaustive) list of the limit laws.
$\lim\limits_{x\to a} x = a$
If $\lim\limits_{x\to a} f(x)$ and $\lim\limits_{x\to a} g(x) = 0$ or $\pm \infty$
Basic laws
$\lim\limits_{x\to a} c = c$ where $c$ is a constant$\lim\limits_{x\to a} x = a$
Addition law
$\lim\limits_{x\to a} (f(x) + g(x)) = \lim\limits_{x\to a} f(x) + \lim\limits_{x\to a} g(x)$
Provided $\lim\limits_{x\to a} f(x)$ and $\lim\limits_{x\to a} g(x)$ exist.
Subtraction law
$\lim\limits_{x\to a} (f(x) - g(x)) = \lim\limits_{x\to a} f(x) - \lim\limits_{x\to a} g(x)$
Provided $\lim\limits_{x\to a} f(x)$ and $\lim\limits_{x\to a} g(x)$ exist.
Multiplication law
$\lim\limits_{x\to a} (f(x) \cdot g(x)) = \lim\limits_{x\to a} f(x) \cdot \lim\limits_{x\to a} g(x)$
Provided $\lim\limits_{x\to a} f(x)$ and $\lim\limits_{x\to a} g(x)$ exist.
Division law
$\lim\limits_{x\to a} \frac{f(x)}{g(x)} = \frac{\lim\limits_{x\to a} f(x)}{\lim\limits_{x\to a} g(x)}$
Provided $\lim\limits_{x\to a} f(x)$ and $\lim\limits_{x\to a} g(x)$ exist and $\lim\limits_{x\to a} g(x) \neq 0$
Power law
$\lim\limits_{x\to a} (f(x))^n = \left( \lim\limits_{x\to a} f(x)) \right)^n$
Provided $\lim\limits_{x\to a} f(x)$ exists and $n$ is an integer.
l'Hôpital's Rule
$\lim\frac{f(x)}{g(x)}=\lim\frac{f'(x)}{g'(x)}$If $\lim\limits_{x\to a} f(x)$ and $\lim\limits_{x\to a} g(x) = 0$ or $\pm \infty$