GRANICA FUNKCJI

$$\operatorname{}{\sqrt[n]{n} = 1}$$

$$\operatorname{}\left( 1 + \frac{1}{n} \right)^{n} = e,\ e = 2,718$$

$$\operatorname{}\sqrt[k]{1 + a_{n} = 1}\text{\ gdy\ kϵN\ i\ lim}a_{n} = 0$$

$$\operatorname{}{\sqrt[n]{a} = 1\ gdy\ a > 0}$$

$$\operatorname{}a^{n} = \left\{ \begin{matrix} 0\ gdy\ \left| a \right| < 1 \\ 1\ gdy\ a = 1 \\ + \infty\ gdy\ a > 1(nie\ istnieje\ a \leq - 1) \\ \end{matrix} \right.\ $$

$$\operatorname{}{\left( 1 + a_{n} \right)^{\frac{1}{a_{n}\ }} = e,\ gdy\ \operatorname{}{a_{n} = 0}\text{\ \ \ \ }}$$

$$\operatorname{}{\left( 1 + b_{n} \right)^{\frac{1}{b_{n}}} = e,\ gdy\ \operatorname{}{b_{n} = 0}\text{\ \ \ \ }}\operatorname{}{a_{n} = 0}\text{\ \ \ }a_{n} > 0\ \ a_{n} < 0$$	$\lim\frac{1}{a_{n}} = \ $ +∞ - ∞
|an| = +∞	$$\operatorname{}{\frac{1}{a_{n}} = 0}$$
$$\left| a_{n} \right| < M\ \frac{\lim b_{n} = 0}{\lim\left| b_{n} \right| = \infty}$$	$$\frac{\lim\left( a_{n}\ \bullet \ b_{n} \right) = 0}{\lim\frac{a_{n}}{b_{n}} = 0}$$
an =   + ∞      b > 0 i limbn = b      b < 0	lim (an•bn) =                        + ∞ -∞
lim|an| = ∞    i    lim bn = 0	lim(an•bn)=
liman = limbn =   + ∞ Albo liman = limbn =   − ∞	lim(an •bn) =   + ∞ lim(an− bn)= $$\lim\frac{a_{n}}{b_{n}} =$$
liman = limbn = 0	$$\lim\frac{a_{n}}{b_{n}} =$$

POCHODNE FUNKCJI

[f(x)∓g(x)]^′ = f′(x) ∓ g′(x)

[f(x)•g(x)]^′ = f^′(x) • g(x) + g^′(x) • f(x)

$$\left\lbrack \frac{\text{f\ }\left( x \right)}{\text{g\ }\left( x \right)} \right\rbrack^{'} = \ \frac{f^{'}\left( x \right)g\left( x \right) - f(x)g'(x)}{g^{2}(x)}$$

[c f(x)]’ = c f’(x) c ϵ R i f(x) jest różniczkowalna

f(x)	f’(x)
f(x)=c	f’(x) = 0
f(x) = x^2	f^′(x) =  α  • x^α − 1
$$f\left( x \right) = \ \sqrt[n]{x}$$	$$f'\left( x \right) = \ \frac{1}{\text{n\ }\sqrt[n]{x^{n - 1}}}$$
f(x) =  a^x	f^′(x) =  a^xlna
f(x) =  e^x	f^′(x) = e^x
f(x) = sinx	f’(x) = cosx
f(x) = cosx	f’(x) = sinx
f(x) =  x	$$f^{'}\left( x \right) = \ \frac{1}{x\ln_{a}}$$
f(x) = lnx	$f^{'}\left( x \right) = \ \frac{1}{x}$
f(x) = arcsinx	$$f^{'}\left( x \right) = \ \frac{1}{\sqrt{1 - x^{2}}}$$
f(x) = arccosx	$$f^{'}\left( x \right) = \ \frac{- 1}{\sqrt{1 - x^{2}}}$$
f(x) = arctgx	$$f^{'}\left( x \right) = \ \frac{1}{x^{2} + 1}$$
f(x) = arcctgx	$$f^{'}\left( x \right) = \ \frac{- 1}{x^{2} + 1}$$
$$f\left( x \right) = \sqrt{g(x)}$$	$$f^{'}\left( x \right) = \ \frac{g'(x)}{2\sqrt{g(x)}}$$
f(x) = ln(g(x))	$$f^{'}\left( x \right) = \ \frac{g'(x)}{g(x)}$$
f(x) =  e^g(x)	f^′(x) =  e^g(x) • g′(x)