\subsubsection {}
Necht' $D(f)={\Bbb R }$ a $a, A\in{\Bbb R }$. Kter\'e z
n\'asleduj\'\i c\'\i ch v\'yrok\accent23 u jsou
ekvivalentn\'\i {} s v\'yrokem $\lim_{x\to a}f(x)=A ?$
$$\text{(a) }\forall_{\epsilon>0}\exists_{K>0}
\exists_{\delta>0}\forall_{x\in\Bbb
R}(x\in P(a,\delta)) \Rightarrow |f(x)-A|<K\epsilon;$$
$$\text{(b) }\exists_{\delta>0}\forall_{\epsilon>0}\forall_{x\in\Bbb
R}(0<|x-a|<\delta)\Rightarrow f(x)\in O(A,\epsilon);$$
(c) Pro ka\v zdou ryze monot\'onn\'\i {} posloupnost $(x_n)$,
pro kterou $\lim_{n\to\infty}x_n=a$, je
$\lim_{n\to\infty}f(x_n)=A.$
 



\subsubsection {}
M\'a $f$ v bod\v e $a$ limitu, jestli\v ze pro ka\v
zdou posloupnost re\'aln\'ych \v c\'\i sel $x_n$, kter\'a
m\'a limitu $a$, plat\'\i {}, \v ze posloupnost $f(x_n)$ m\'a
limitu?
 



\subsubsection {}
Jak\'e jsou n\'asleduj\'\i c\'\i {} limity?
$$\lim_{x\to\infty}\frac{x}{[x]}, \lim_{x\to 1}([x]-x),
\lim_{x\to 0} x\cdot [\frac{1}{x}].$$
 



\subsubsection {}
Zkoumejte limity func\'\i {} : Dirichletova funkce,
Riemannova funkce, $[x], \{x\}, \operatorname{sign} x$ ve
v\v sech bodech ${\Bbb R }$.
 



\subsubsection {}
Existuje funkce $f$ zobrazuj\'\i c\'\i {} ${\Bbb R }$ do
${\Bbb R }$, kter\'a je spojit\'a v nule a nespojit\'a v $a$,
pokud $a\in{\Bbb R }\setminus\{0\}$?

Existuje funkce $g$ zobrazuj\'\i c\'\i {} ${\Bbb R }$ do $\Bbb
R$, kter\'a je spojit\'a v $a$, pr\'av\v e kdy\v z $a \ne
2$?
 



\subsubsection {}
}$$\lim_{x\to a}\frac{3x^2+1}{2x^2+1}, \lim_{x\to
a}\frac{2x^2+1}{x^2-1}, \lim_{x\to a}\frac{2x^3+1}{x^3+1}$$
pro v\v sechny hodnoty $a\in{\Bbb R }^*$.
 



\subsubsection {}
Po\v c\'\i tejte n\'asleduj\'\i c\'\i {} limity funkc\'\i {}.

\c{67} $$\lim_{x\to a}\frac{x^2-1}{2x^2-x-1},\text{ pokud
}a=0, 1, \infty, -\infty.$$


\subsubsection {}
$$\lim_{x\to 1}\frac{x^3-3x+2}{x^4-4x+3}, \lim_{x\to
2}\frac{(x^2-x-2)^{20}}{(x^3-12x+16)^{10}}, \lim_{x\to
1}\frac{x^2-1}{x^3-x^2-x+1}.$$
 



\subsubsection {}
$$\lim_{x\to 1}\frac{x^m-1}{x^n-1}, \lim_{x\to
1}\frac{x+x^2+\dots+x^n-n}{x-1},$$
$$\lim_{x\to 0}\frac{(1+mx)^n-(1+nx)^m}{x^2}, \lim_{x\to
1}(\frac{m}{1-x^m}-\frac{n}{1-x^n}).$$
 



\subsubsection {}
$$\lim_{x\to 0}\frac{\sqrt{1+x}-\sqrt{1-x}}{x},
\lim_{x\to 7}\frac{\sqrt{x+2}-\root 3\of {x+20}}{\root
4\of{x+9}-2}.$$
 



\subsubsection {}
$$\lim_{x\to 0}\frac{\root n \of {1+x}-1}{x},
\lim_{x\to 1}\frac{\root m\of {x} -1}{\root n \of{x}-1}.$$
 



\subsubsection {}
$$\lim_{x\to\infty} \{\sqrt{(x+a)(x+b)}-x\},
\lim_{x\to\infty}\{\root k \of
{(x+a_1)\cdot(x+a_2)\cdot\dots\cdot(x+a_k)}-x\}.$$
 



\subsubsection {}
Najd\v ete $a,b\in{\Bbb R }$ tak, aby platilo
$$\lim_{x\to-\infty}(\sqrt{x^2-x+1}-ax-b) = 0.$$

Zkuste zn\'azornit grafy funkc\'\i {} $\sqrt{x^2-x+1}$ a
$ax+b$ a posoudit, co obdr\v zen\'y v\'ysledek znamen\'a.
 



\subsubsection {}
$$\lim_{x\to\infty}x(\sqrt{x^2+2x}-2\sqrt{x^2+x}+x).$$
 



\subsubsection {}
$$\lim_{x\to 0}\frac{\root m \of {1+ax}-\root n
\of{1+bx}}{x}.$$
 



\subsubsection {}
Spo�t�te limity
$$\lim\limits_{x\to 0}\frac{\cos x - \cos 3x}{x^2},
  \lim\limits_{x\to 0}\frac{\cos mx - \cos nx}{x^2}.$$

 



\subsubsection {}
Spo�t�te limity
$$\lim\limits_{x\to 0}\frac{\sinh (x)}{x},
  \lim\limits_{x\to 0}\frac{\tanh (x)}{x}.$$
 



\subsubsection {}
Doka�te nebo odvo�te rovnosti
$$\arcsin x + \arccos x = ... ?,
  \arctan x + \arctan (\frac{1}{x}) = ... ,$$
$$\arctan x = \arcsin \frac{x}{\sqrt{1+x^2}},
  \arctan x + \arctan y = \arctan \frac{x + y}{1 - xy},$$
maj�-li ob� strany smysl. Jak je tomu ve zbyl�ch p��padech?
 



\subsubsection {}
Spo�t�te limity
$$\lim\limits_{x\to \pi}\frac{\sin ax}{\sin bx},
  \lim\limits_{x\to 0}\frac{1+\sin x - \cos x}{1+\sin px -
  \cos px}, p\in{\Bbb R },$$
$$\lim\limits_{x\to \frac{\pi}{4}}\tan 2x \tan
(\frac{\pi}{4}- x).$$
 



\subsubsection {}
Po��tejte limity
$$\lim\limits_{x\to 0}\frac{\sin(\sin(\sin x))}
{\cos(\frac{\pi}{2}\cos x)}\cdot x^k,
  \lim\limits_{x\to 0}\frac{\sin(\sin x - x^2)}{x},$$




\subsubsection {}
Spo�t�te $\lim\limits_{x\to \infty}
(\cos\sqrt{x+1}-\cos\sqrt{x-1}).$
 



\subsubsection {}
Zat�m bez d�kazu pou��v�me $\sin x \le x$. Spo�t�te
$$\lim\limits_{n\to \infty}a_n,$$
kde $a_{n+1}=\sin(a_n)$ a $a_1=\sin x$. 



\subsubsection {}
Zformulujte a doka�te analogii Heineho v�ty pro
jednostrann� limity. Lze jednostrann� limity definovat t�
s pou�it�m monot�nn�ch posloupnost�?
 



\subsubsection {}
Spo�t�te
$$\lim\limits_{0_+}\frac{\sin^2\sqrt{x}\tan(\sin
x)}{\cos x - 1}, \lim\limits_{0_+}\frac{e^{\sqrt{\sin
x}}-\cos x}{\sqrt x},$$
$$\lim\limits_{x\to 0}\left(\frac{1+\tan x}{1+\sin
x}\right)^{\frac{1}{\sin ^3 x}}.$$
 



\subsubsection {}
S u�it�m zn�m�ch vlastnost� exponenci�ln� funkce
$\exp{x}=e^x$ spo�t�te
$$\lim\limits_{x\to
\infty}\left(\frac{x^2+1}{x^2-2}\right)^{x^2},
\   
\lim\limits_{x\to  0}\left(\frac{1+x2^x}{1+x3^x}\right)^{\frac{1}{x^2}}.$$
 



\subsubsection {}
Znalosti logaritmu pou�ijte k v�po�tu
$$\lim\limits_{x\to\infty}\frac{\log(x^2-x+1)}{\log(x^{10}+x+1)},
  \lim\limits_{x\to 0}\frac{\log\cos ax}{\log\cos bx},$$
$$\lim\limits_{x\to 0_+}\log(x\log a)\cdot
\log\left(\frac{\log ax}{\log{\frac{x}{a}}}\right), a>1.$$
 



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



\subsubsection {}
S pomoc� tvrzen�, �e
$\lim\limits_{n\to\infty}\frac{n}{e^n}=0$, kter� bylo
dok�z�no na p�edn�ce (zopakujte si!), doka�te, �e
$$\lim\limits_{x\to\infty}\frac{x^a}{e^x}=0, a\in{\Bbb R }.$$

 



\subsubsection {}
Podobn� zkoumejte
$$\lim\limits_{x\to\infty}x^a\cdot e^{bx},
\lim\limits_{y\to\infty}y^a\cdot\log y^b$$
pro r�zn� $a,b$ re�ln�. 



\subsubsection {}
 S pou�it�m p�edchoz�ho spo��tejte
$$\lim\limits_{n\to\infty}\log(1+2^x)\log(1+\frac{3}{x}),
  \lim\limits_{n\to\infty}\left(\frac{1}{n^2}\right)^{\frac{2n}{n+1}},$$
$$\lim\limits_{n\to\infty}\left(\frac{n}{n+1}\right)^n,
  \lim\limits_{n\to\infty}\left(\frac{n^n-1}{n^n+3}\right)^{n^n+2},$$
$$\lim\limits_{n\to\infty}(1-\frac{2}{n})^n.$$
 

