User:Foxjwill/Limit Notes

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[edit] 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)}

[edit] 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}

[edit] 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)

[edit] 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.