Limit laws reference

Here is a (non-exhaustive) list of the limit laws.

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 $\pm \infty$