User:Foxjwill/Limit Notes

Limit laws
$$ \lim_{x\rightarrow a} \left [ f(x) + g(x) \right] = \lim_{x\rightarrow a} f(x) + \lim_{x\rightarrow a} g(x)$$ $$ \lim_{x\rightarrow a} \left [ f(x) - g(x) \right] = \lim_{x\rightarrow a} f(x) - \lim_{x\rightarrow a} g(x)$$ $$ \lim_{x\rightarrow a} \left [ cf(x)\right] = c\lim_{x\rightarrow a} f(x)$$ $$ \lim_{x\rightarrow a} \left [ f(x)g(x) \right] = \lim_{x\rightarrow a} f(x)\cdot\lim_{x\rightarrow a} g(x)$$ $$ \lim_{x\rightarrow a} \left [ \frac{f(x)}{g(x)} \right] = \frac{\lim_{x\rightarrow a} f(x)}{\lim_{x\rightarrow a} g(x)}\mbox{ if }\lim_{x\rightarrow a}  g(x) \neq 0$$ $$ \lim_{x\rightarrow a} \left [ f(x) \right]^n = \left [ \lim_{x\rightarrow a} f(x) \right ]^n$$ $$ \lim_{x\rightarrow a} \sqrt[n]{f(x)} = \sqrt[n]{\lim_{x\rightarrow a} f(x)}$$

Special limits
$$\lim_{x\rightarrow a} c = c$$ $$\lim_{x\rightarrow a} x = a$$ $$\lim_{x\rightarrow a} x^n = a^n$$ $$\lim_{x\rightarrow a} \sqrt[n]{x} = \sqrt[n]{a}$$

Limits by direct substitution
If $$f\!$$ is a polynomial or a rational function and $$a\!$$ is in the domain of $$f\!$$, then $$\lim_{x\rightarrow a} f(x) = f(a)$$

Derivitive
The derivative of a function $$f\!$$ at a number $$a\!$$, denoted by $$f'(a)\!$$ is $$f'(a) = \lim_{h\rightarrow 0} \frac{f(a+h) - f(a)}{h}$$ if this limit exists.