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\Nadpis{\bf 10. cvi�en� z MA---5. a 6.12.2005}

\nadpis{Limity funkc�}

Jak je limita funkce definovan�? Jak se li�� limita funkce a limita
posloupnosti? Kter� v�ta je spojuje? Jak� metody zn�te na v�po�et 
limit funkc�? (VOAL, v�ta o policajtech, l'Hospitalovo pravidlo,
Taylor�v polynom)

\bigskip

Spo��tejte n�sleduj�c� limity, nebo doka�te, �e neexistuj�:

\pr $\limx{a}\frac{x^2-1}{2x^2-x-1}$, pro $a=0, 1, \infty, -\infty$



\pr 
\cast $\limx{1}\frac{x^3-3x+2}{x^4-4x+3}$,
\cast $\limx{2}\frac{(x^2-x-2)^{20}}{(x^3-12x+16)^{10}}$.

\pr
\cast $\limx{1}\frac{x^2-1}{x^3-x^2-x+1}$, 
\cast $\limx{1}\frac{x^m-1}{x^n-1}$,
\cast $\limx{1}\frac{x+x^2+\dots+x^n-n}{x-1}$,
\cast $\limx{0}\frac{(1+mx)^n-(1+nx)^m}{x^2}$,
\cast $\limx{1}(\frac{m}{1-x^m}-\frac{n}{1-x^n})$.


\pr
\cast $\limx{\infty}\frac{x}{\floor{x}}$,
\cast $\limx{1}(\floor{x}-x)$,
\cast $\limx{0} x\cdot \floor{\frac{1}{x}}$.

\pr
\cast $\limx{0}\frac{\sqrt{1+x}-\sqrt{1-x}}{x}$,
\cast $\limx{7}\frac{\sqrt{x+2}-\root 3\of {x+20}}{\root 4\of{x+9}-2}$.

\pr
\cast $\limx{0}\frac{\root n \of {1+x}-1}{x}$,
\cast $\limx{1}\frac{\root m\of {x} -1}{\root n \of{x}-1}$.

\pr
\cast $\limx{\infty} \bigl(\sqrt{(x+a)(x+b)}-x\bigr)$,
\cast $\limx{\infty}\bigl(\root k \of {(x+a_1)(x+a_2)\dots(x+a_k)}-x\bigr)$.

\pr
\cast $\limx{\infty}x\bigl(\sqrt{x^2+2x}-2\sqrt{x^2+x}+x\bigr)$,
\cast $\limx{0}\frac{\root m \of {1+ax}-\root\of{1+bx}}{x}$.

\pr Najd�te $a,b\in\er$ tak, aby platilo
$$
  \limx{-\infty}(\sqrt{x^2-x+1}-ax-b) = 0\,.
$$

\oddel

\pr 
\cast $\limx{0}\frac{\sin 5x - \sin 3x}{\sin x}$,
\cast $\limx{0}\frac{1-\cos x}{x^2}$,
\cast $\limx{0}\frac{\cos x - \cos 3x}{x^2}$,
\cast $\limx{0}\frac{\cos mx - \cos nx}{x^2}$
\cast $\limx{0}\frac{\sin(\sin(\sin x))}{\cos(\frac{\pi}{2}\cos x)} x^k$,
\cast $\limx{0}\frac{\sin(\sin x - x^2)}{x}$,
\cast $\limx{0}\frac{\sin(\sin x - x + \frac{x^3}{6})}{x}$,
\cast $\limx{\infty} (\cos\sqrt{x+1}-\cos\sqrt{x-1})$.

\pr
S vyu�it�m vlastnost� exponenci�ln� a logaritmick� funkce spo��tejte
n�sleduj�c� limity

\cast $\limx{0}\frac{\sinh (x)}{x}$,
\cast $ \limx{\infty}\left(\frac{x^2+1}{x^2-2}\right)^{x^2}$,
\cast $\limx{\infty}\frac{\log(1 + \sqrt{x} + \root 3 \of{x})}
     {\log(1  + \root 3 \of{x} + \root 4 \of{x} )}$,
\cast $\limx{\infty}\frac{\log(x^2-x+1)}{\log(x^{10}+x+1)}$,

\oddel

\pr
Rozhodn�te, pro kter� $a\in \er$ lze funkci
$$
  f(x)=x^a(\arcsin x- \text{arctg} x)
$$
spojit� roz���it na $\er$. Tot� zjist�te pro funkci
$$
  g(x)=x^a-\sin(\log x)\cdot\text{arctg} x \,.
$$

\oddel

\pr Spo�t�te limity posloupnost�. 
\cast $\lp n^2 \Biggl( \log \bigl(1 + \frac 1n\bigr) - \sin \frac 1n\Biggr)$,
\cast $\lp n^8 \Biggl( 2 \cos \frac 1{n^2} - 3 + \sqrt{1 + \frac 2{n^4}}\Biggr)$,
\cast $\lp \frac { \log (n^{-2} + e^{1/n}) }{ \log (n^{-4} + e^{2/n}) }$.


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