\magnification=1200
\NoBlackBoxes
\documentstyle{amsppt}
\document



{\bf Re�ln� funkce, pojem limity a spojitosti v bod�}
\medskip
\medskip
Probl�m 1:

Bu� $f: X\to Y$ zobrazen� a ozna�me $D_f$ defini�n� obor $f$,  obraz mno�iny $A \subset X$  a $f^{-1}(B)$ vzor mno�iny $B\subset Y$, tj.$f(A)$
$$
\align\split
&D_f = \{x \in X; \exists y \in Y \text{tak, �e} f(x)= y\},\\
&f(A) = \{y \in Y; \exists x \in A \text{tak, �e} f(x)= y\},\\
&f^{-1}(B) = \{x \in X; \exists y \in B \text{tak, �e} f(x)= y\}.
\endsplit
\endalign
$$

Rozhodn�te, zda plat� resp. za jak�ch p�edpoklad� plat�
$\forall A \subset X : f^{-1}(f(A)) = A $, $\forall B \subset Y: f(f^{-1}(B)) = B$.
\medskip
Probl�m 2:

Doka�te, �e funkce $f$ je na intervalu $I$
rostouc�, pr�v� kdy� plat�
$$\forall \ \ x,y \in I, x \ne y: \frac{f(x)-f(y)}{x-y}>0.$$
\medskip
Probl�m 3:

Je-li $f$ neklesaj�c� na $(-\infty,a>$ a
nerostouc� na $<a,\infty)$ pro n�jak� $a \in \Bbb
R$, pak $f$ nab�v� maxima.
\medskip
Probl�m 4:

Nech� $f$ nab�v� minima v $a\in \Bbb R$. Mus�
pak existovat $\epsilon>0$ takov\'e, \v ze $f$ je
nerostouc� na $(a-\epsilon,a>$ a neklesaj�c�
na $<a,a+\epsilon)$?
\medskip
Probl�m 5:

Kter� z n�sleduj�c�ch operac�
proveden�ch na neklesaj�c� (monotonn�) funkce d�vaj�
op�t neklesaj�c� (monotonn�) funkci?

Operace: $+, -,\max, \min, \circ.$

Existuje funkce, kter� nen�  monotonn� na 
��dn�m intervalu?
\medskip
P��klad 1:

Studujte vlastnosti a nakreslete grafy
n\'asleduj\'\i c\'\i ch funkc\'\i {}:

$$f_1(x)=\frac{2x}{x^2+1}, f_2(x)=x^n (n\in\Bbb N),
f_3(x)=\frac{5-|x-7|}{2},$$
$$f_4(x)=(x-5)^2+6, f_5(x)=x^2+2x-7, f_6(x)=\frac{-1}{|x-3|}.$$
\medskip
P��klad 2:

Spo��tejte limity nebo doka�te, �e neexistuj�
$$
\align
\split
\lim_{x\to a}\frac{x^2-1}{2x^2-x-1},\text{ pokud
}a=0, 1, \infty, -\infty; \lim_{x\to 1}\frac{x^3-3x+2}{x^4-4x+3}, \lim_{x\to
2}\frac{(x^2-x-2)^{20}}{(x^3-12x+16)^{10}},\\
 \lim_{x\to1}\frac{x^2-1}{x^3-x^2-x+1}, \lim_{x\to 1}\frac{x^m-1}{x^n-1}, \lim_{x\to
1}\frac{x+x^2+\dots+x^n-n}{x-1},\\
\lim_{x\to 0}\frac{(1+mx)^n-(1+nx)^m}{x^2}, \lim_{x\to
1}(\frac{m}{1-x^m}-\frac{n}{1-x^n}).
\endsplit
\endalign
$$

$$
\lim_{x\to 0}\frac{\sqrt{1+x}-\sqrt{1-x}}{x},
\lim_{x\to 7}\frac{\sqrt{x+2}-\root 3\of {x+20}}{\root
4\of{x+9}-2}.
$$
$$
\lim_{x\to 0}\frac{\root n \of {1+x}-1}{x},
\lim_{x\to 1}\frac{\root m\of {x} -1}{\root n \of{x}-1}.
$$
$$
\lim_{x\to\infty} \{\sqrt{(x+a)(x+b)}-x\},
\lim_{x\to\infty}\{\root k \of
{(x+a_1)\cdot(x+a_2)\cdot\dots\cdot(x+a_k)}-x\}.
$$
$$
\lim_{x\to\infty}x(\sqrt{x^2+2x}-2\sqrt{x^2+x}+x), \lim_{x\to 0}\frac{\root m \of {1+ax}-\root n
\of{1+bx}}{x}.
$$
\medskip
P��klad 3:

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

\medskip
P��klad 4:

Spo��tejte limity nebo doka�te, �e neexistuj�
$$\lim\limits_{x\to 0}\frac{\sin 5x - \sin 3x}{\sin x},
  \lim\limits_{x\to 0}\frac{1-\cos x}{x^2},
  $$
$$\lim\limits_{x\to 0}\frac{\cos x - \cos 3x}{x^2},
  \lim\limits_{x\to 0}\frac{\cos mx - \cos nx}{x^2}.
  $$
$$\lim\limits_{x\to \pi}\frac{\sin ax}{\sin bx},
  \lim\limits_{x\to 0}\frac{1+\sin x - \cos x}{1+\sin px -
  \cos px}, a,b,p\in\Bbb R, b\ne 0, p\ne 0
  $$
$$\lim\limits_{x\to \frac{\pi}{4}}\text{tg} 2x \text{tg}
(\frac{\pi}{4}- x), 
$$
$$\lim\limits_{x\to 0}\frac{\sin(\sin(\sin x))}{\cos(\frac{\pi}{2}\cos x)} x^k,
  \lim\limits_{x\to 0}\frac{\sin(\sin x - x^2)}{x},
  $$
$$\lim\limits_{x\to 0}\frac{\sin(\sin x - x +
\frac{x^3}{6})}{x}.
$$
$$ 
\lim\limits_{x\to \infty}
(\cos\sqrt{x+1}-\cos\sqrt{x-1}).
$$
$$
\lim\limits_{x\to 0_+}\frac{\sin^2\sqrt{x}\text{tg}(\sin
x)}{\cos x - 1}, \lim\limits_{x\to 0_+}\frac{e^{\sqrt{\sin
x}}-\cos x}{\sqrt x},
$$
$$
\lim\limits_{x\to 0}\left(\frac{1+\text{tg} x}{1+\sin
x}\right)^{\frac{1}{\sin ^3 x}},
$$

\medskip
P��klad 5:

S vyu�it�m vlastnost� exponenci�ln� a logaritmick� funkce spo��tejte 
n�sleduj�c� limity
$$
\lim\limits_{x\to 0}\frac{\sinh (x)}{x},
  \lim\limits_{x\to 0}\frac{\text{tgh} (x)}{x},
  $$

$$
\lim\limits_{x\to
\infty}\left(\frac{x^2+1}{x^2-2}\right)^{x^2},
\lim\limits_{x\to  0}(1 + x^2)^{\text{cotg}^2 x},   
\lim\limits_{x\to  0}\left(\frac{1+x2^x}{1+x3^x}\right)^{\frac{1}{x^2}},
$$
$$
\lim\limits_{x\to  0}(\frac{1 + \text{tg} x}{1 + \sin x})^{\frac{1}{\sin^2 x}},
\lim\limits_{x\to  0}x \sqrt{|\cos\frac{1}{x}|},
\lim\limits_{x\to  0}(\frac{a^x + b^x + c^x}{3})^{\frac{1}{x}},
$$
$$
\lim\limits_{x\to  \infty}\frac{\log(1 + \sqrt{x} + \root 3 \of{x} )}{\log(1  + \root 3 \of{x} + \root 4 \of{x} )},
\lim\limits_{x\to  0}\log (1 + 2^x) \log(1 + \frac{3}{x}),
$$
$$\lim\limits_{x\to\infty}\frac{\log(x^2-x+1)}{\log(x^{10}+x+1)},
  \lim\limits_{x\to 0}\frac{\log\cos ax}{\log\cos bx},
  $$
$$\lim\limits_{x\to 0_+}\log(x\log a)\cdot
\log\left(\frac{\log ax}{\log{\frac{x}{a}}}\right), a>1.$$
\medskip
P��klad 6:

Spo��tejte limity nebo doka�te, �e neexistuj�
$$\lim\limits_{x\to \infty}x(\frac{\pi}{2} - \arcsin \frac{x}{\sqrt{x^2 + 1}}),
\lim\limits_{x\to 0}\frac{\text{arctg}(1 + x)- \text{arctg}(1-x)}{x}.
$$
\medskip

P��klad 7:

Spo��tejte limity posloupnost�
$$\lim\limits_{n\to\infty}\log(1+2^n)\log(1+\frac{3}{n}),
  \lim\limits_{n\to\infty}\left(\frac{1}{n^2}\right)^{\frac{2n}{n+1}},$$
$$\lim\limits_{n\to\infty}\left(\frac{n}{n+1}\right)^n,
  \lim\limits_{n\to\infty}\left(\frac{n^n-1}{n^n+3}\right)^{n^n+2},$$
$$\lim\limits_{n\to\infty}(1-\frac{2}{n})^n.$$
$$\lim\limits_{n\to \infty}a_n,$$
kde $a_{n+1}=\sin(a_n)$ a $a_1=\sin x$.
\medskip
Probl�m 6:

Rozhodn�te, pro kter� $a\in \Bbb R$ lze funkci
$$f(x)=x^a(\arcsin x- \text{arctg} x)$$
spojit� roz���it na $\Bbb R$. Tot� zjist�te pro funkci
$$g(x)=x^a-\sin(\log x)\cdot\text{arctg} x.
$$
\medskip
Probl�m 7:

Jak� jsou n�sleduj�c� limity?
$$\lim_{x\to\infty}\frac{x}{[x]}, \lim_{x\to 1}([x]-x),
\lim_{x\to 0} x\cdot [\frac{1}{x}].$$

Zkoumejte limity func�: Dirichletova funkce,
Riemannova funkce, $[x], \{x\}, \operatorname{sign} x$ ve
v�ech bodech $\Bbb R$.
\medskip
Probl�m 8:

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

Existuje funkce $g$ zobrazuj�c� $\Bbb R$ do $\Bbb
R$, kter� je spojit� v $a$, pr�v� kdy� $a \ne
2$?

\medskip
P��klad 9:

U n�sleduj�c�ch funkc� ur�ete defini�n� obor, body spojitosti, limity v krajn�ch bodech defini�n�ho oboru resp. v bodech nespojitosti a spo��tejte derivaci
$$
x^x, x^{\frac{1}{x}}, (\sin x)^{\cos x}, \text{arctg}(\text{tg}^2 x), \arcsin(\sin^2 x), 
\log(\arccos x),
$$
$$
 x \arcsin\sqrt{\frac{x}{x+1}} + \text{arctg }\sqrt{x - \sqrt{x}},
\text{arctg}(e^x) - \log\sqrt{\frac{e^{2x}}{e^{2x}+ 1}}.
$$


\bye