%% This document created by Scientific Notebook (R) Version 3.5 %% Starting shell: article \documentclass{article} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \usepackage{amssymb} %TCIDATA{OutputFilter=LATEX.DLL} %TCIDATA{Version=5.00.0.2570} %TCIDATA{} %TCIDATA{Created=Wednesday, February 10, 1999 13:29:48} %TCIDATA{LastRevised=Sunday, February 13, 2005 16:50:21} %TCIDATA{} %TCIDATA{} %TCIDATA{CSTFile=On line bluem.cst} %TCIDATA{PageSetup=72,72,72,72,0} %TCIDATA{Counters=arabic,1} %TCIDATA{AllPages= %H=36 %F=36,\PARA{038
\QTR{small}{Matematick\U{e1} anal\U{fd}za II online - Funkcie, limita funkcie, spojit\U{e9} funkcie - Limita funkcie\dotfill \thepage }} %} \newtheorem{theorem}{Theorem} \newtheorem{acknowledgement}[theorem]{Acknowledgement} \newtheorem{algorithm}[theorem]{Algorithm} \newtheorem{axiom}[theorem]{Axiom} \newtheorem{case}[theorem]{Case} \newtheorem{claim}[theorem]{Claim} \newtheorem{conclusion}[theorem]{Conclusion} \newtheorem{condition}[theorem]{Condition} \newtheorem{conjecture}[theorem]{Conjecture} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{criterion}[theorem]{Criterion} \newtheorem{definition}[theorem]{Definition} \newtheorem{example}[theorem]{Example} \newtheorem{exercise}[theorem]{Exercise} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{notation}[theorem]{Notation} \newtheorem{problem}[theorem]{Problem} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{remark}[theorem]{Remark} \newtheorem{solution}[theorem]{Solution} \newtheorem{summary}[theorem]{Summary} \newenvironment{proof}[1][Proof]{\textbf{#1.} }{\ \rule{0.5em}{0.5em}} \input{tcilatex} \begin{document} \author{A. U. Thor} \title{Lab Report} \date{The Date } \maketitle \begin{abstract} A Laboratory report created with Scientific Notebook \end{abstract} \section{Funkcie, limita funkcie, spojit\'{e} funkcie} \begin{center} \begin{tabular}{|c|c|c|c|c|c|c|} \hline \textbf{% %TCIMACRO{\hyperref{Obsah}{}{}{maiindex.tex}}% %BeginExpansion \msihyperref{Obsah}{}{}{maiindex.tex}% %EndExpansion } & \textbf{% %TCIMACRO{\hyperref{Obsah kapitoly}{}{}{Ma2.tex}}% %BeginExpansion \msihyperref{Obsah kapitoly}{}{}{Ma2.tex}% %EndExpansion } & \textbf{% %TCIMACRO{\hyperref{Predch\'{a}dzaj\'{u}ca str\'{a}nka}{}{}{Ma21.tex}}% %BeginExpansion \msihyperref{Predch\'{a}dzaj\'{u}ca str\'{a}nka}{}{}{Ma21.tex}% %EndExpansion } & \textbf{% %TCIMACRO{\hyperref{Nasleduj\'{u}ca str\'{a}nka}{}{}{Ma23.tex}}% %BeginExpansion \msihyperref{Nasleduj\'{u}ca str\'{a}nka}{}{}{Ma23.tex}% %EndExpansion } & \textbf{% %TCIMACRO{\hyperref{Ot\'{a}zky}{}{}{O2.tex}}% %BeginExpansion \msihyperref{Ot\'{a}zky}{}{}{O2.tex}% %EndExpansion } & \textbf{% %TCIMACRO{\hyperref{Cvi\v{c}enia}{}{}{C2.tex}}% %BeginExpansion \msihyperref{Cvi\v{c}enia}{}{}{C2.tex}% %EndExpansion } & \textbf{% %TCIMACRO{\hyperref{Index}{}{}{Glos.tex}}% %BeginExpansion \msihyperref{Index}{}{}{Glos.tex}% %EndExpansion } \\ \hline \end{tabular} \end{center} \subsection{Limita funkcie.} Analogick\'{y}m sp\^{o}sobom ako v matematickej anal\'{y}ze I zavedieme pojem limity funkcie viacer\'{y}ch premenn\'{y}ch v bode. \begin{definition} Nech $\mathbf{f}:A\left( \subset \mathbf{R}^{n}\right) \longrightarrow \mathbf{R}^{m}.$ Ak $\mathbf{a}\in \mathbf{R}^{n}$ je hromadn\'{y}m bodom mno% \v{z}iny $A,$ hovor\'{\i}me, \v{z}e $\label{11}$\emph{limita} $\mathbf{f}(% \mathbf{x}),$ \emph{ak sa} $\mathbf{x}$ \emph{bl\'{\i}\v{z}i ku} $\mathbf{a}$ \emph{sa rovn\'{a}} $\mathbf{b}\in \mathbf{R}^{m}$ a zapisujeme $\lim_{% \mathbf{x}\longrightarrow \mathbf{a}}\mathbf{f}\left( \mathbf{x}\right) =% \mathbf{b}$ vtedy a len vtedy ak $\forall O_{\varepsilon }\left( \mathbf{b}% \right) \,\exists O_{\delta }^{\circ }\left( \mathbf{a}\right) ;\,\mathbf{f}% \left( O_{\delta }^{\circ }\left( \mathbf{a}\right) \cap A\right) \subset O_{\varepsilon }\left( \mathbf{b}\right) .$ \end{definition} T\'{u}to defin\'{\i}ciu m\^{o}\v{z}me prep\'{\i}sa\v{t} pomocou nerovnost% \'{\i} do nasleduj\'{u}ceho tvaru \[ \forall \varepsilon >0\,\,\exists \delta >0;\,\mathbf{x}\in A\wedge \,0<\left\| \mathbf{x-a}\right\| <\delta \Longrightarrow 0<\left\| \mathbf{f}% \left( \mathbf{x}\right) \mathbf{-b}\right\| <\varepsilon . \] \begin{description} \item[Pozn\'{a}mka] V predch\'{a}dzaj\'{u}cej defin\'{\i}cii sme \v{z}% iadali, aby bod $\mathbf{a}$ bol hromadn\'{y}m bodom mno\v{z}iny $A,$ to znamen\'{a}, \v{z}e bod $\mathbf{a}$ nemus\'{\i} by\v{t} bodom mno\v{z}iny $% A.$Defin\'{\i}ciu limity funkcie v bode budeme pou\v{z}\'{\i}va\v{t} oboch formul\'{a}ci\'{a}ch, pomocou okol\'{\i} ako aj s \v{n}ou ekvivalentn\'{u} defin\'{\i}ciu pomocou nerovnost\'{\i}. \end{description} \begin{example} \label{8}Nech $f:\mathbf{R}^{n}\longrightarrow \mathbf{R},\,f\left( \mathbf{x% }\right) =\left\| \mathbf{x}\right\| .$ Potom $\lim_{\mathbf{x}% \longrightarrow \mathbf{a}}\left\| \mathbf{x}\right\| =\left\| \mathbf{a}% \right\| .$ \end{example} \begin{solution} Plat\'{\i} \[ \left\| f\left( \mathbf{x}\right) \mathbf{-}f\left( \mathbf{a}\right) \right\| =\left| f\left( \mathbf{x}\right) \mathbf{-}f\left( \mathbf{a}% \right) \right| =\left| \left\| \mathbf{x}\right\| -\left\| \mathbf{a}% \right\| \right| \leq \left\| \mathbf{x-a}\right\| , \] teda ak v defin\'{\i}cii limity funkcie zvol\'{\i}me $\varepsilon >0$ a $% \delta =\varepsilon ,$ potom ak \[ \left\| \mathbf{x-a}\right\| <\delta \Longrightarrow \left| \left\| \mathbf{x% }\right\| -\left\| \mathbf{a}\right\| \right| =\left\| f\left( \mathbf{x}% \right) \mathbf{-}f\left( \mathbf{a}\right) \right\| <\varepsilon , \] t.j. $\lim_{\mathbf{x}\longrightarrow \mathbf{a}}\left\| \mathbf{x}\right\| =\left\| \mathbf{a}\right\| .\square $ \end{solution} \begin{example} \label{9}Nech $\pi _{i}:\mathbf{R}^{n}\longrightarrow \mathbf{R},\,\pi _{i}\left( \mathbf{x}\right) =x_{i},\,i=1,2,...,n$ t.j. funkcia $\pi _{i}$ je projekcia na i-tu komponentu prvku $\mathbf{x.}$ Potom $\lim_{\mathbf{x}% \longrightarrow \mathbf{a}}\pi _{i}\left( \mathbf{x}\right) =a_{i}.$ \end{example} \begin{solution} Plat\'{\i} $\left\| \pi _{i}\left( \mathbf{x}\right) \mathbf{-}\pi _{i}\left( \mathbf{a}\right) \right\| =\left| \pi _{i}\left( \mathbf{x}% \right) \mathbf{-}\pi _{i}\left( \mathbf{a}\right) \right| =\left| x_{i}-a_{i}\right| \leq \left\| \mathbf{x-a}\right\| ,$ teda ak v defin\'{\i}% cii limity funkcie zvol\'{\i}me $\varepsilon >0$ a $\delta =\varepsilon ,$ potom ak \[ \left\| \mathbf{x-a}\right\| <\delta \Longrightarrow \left| x_{i}-a_{i}\right| =\left\| \pi _{i}\left( \mathbf{x}\right) \mathbf{-}\pi _{i}\left( \mathbf{a}\right) \right\| <\varepsilon , \] t.j. $\lim_{\mathbf{x}\longrightarrow \mathbf{a}}\pi _{i}\left( \mathbf{x}% \right) =a_{i}.\square $ \end{solution} \begin{lemma} \label{15}Nech $\mathbf{f}:A\left( \subset \mathbf{R}^{n}\right) \longrightarrow \mathbf{R}^{m},\,C\subset A.$ Nech $\mathbf{a}$ je hromadn% \'{y}m bodom mno\v{z}iny $A$ aj mno\v{z}iny $C.$Nech $\mathbf{g=f|}_{C}.$ Potom ak $\lim_{\mathbf{x}\longrightarrow \mathbf{a}}\mathbf{f}\left( \mathbf{x}\right) =\mathbf{b,}$ tak aj $\lim_{\mathbf{x}\longrightarrow \mathbf{a}}\mathbf{g}\left( \mathbf{x}\right) =\mathbf{b}$ a limitu $\lim_{% \mathbf{x}\longrightarrow \mathbf{a}}\mathbf{g}\left( \mathbf{x}\right) =% \mathbf{b}$ naz\'{y}vame limitou funkcie $\mathbf{f}$ \emph{vzh\v{l}adom na mno\v{z}inu} $C$ a p\'{\i}\v{s}eme $\lim_{\mathbf{x}\longrightarrow \mathbf{% a,x}\in C}\mathbf{f}\left( \mathbf{x}\right) =\mathbf{b}.$ \end{lemma} \begin{tabular}{|c|} \hline {\small %TCIMACRO{\hyperref{D\^{o}kaz}{}{}{DO211.tex}}% %BeginExpansion \msihyperref{D\^{o}kaz}{}{}{DO211.tex}% %EndExpansion } \\ \hline \end{tabular}% \label{1} \begin{description} \item[Pozn\'{a}mka] \label{10}Ak existuje $\lim_{\mathbf{x}\longrightarrow \mathbf{a}}\mathbf{f}\left( \mathbf{x}\right) =\mathbf{b},$ tak \textsl{pre ka\v{z}d\'{u} podmno\v{z}inu} $M\subset A$ defini\v{c}n\'{e}ho oboru funkcie $\mathbf{f},$ \textsl{ktorej hromadn\'{y}m bodom je bod} $\mathbf{a},$ plat% \'{\i} \ $\lim_{\mathbf{x}\longrightarrow \mathbf{a,x}\in M}\mathbf{f}\left( \mathbf{x}\right) =\mathbf{b}.$ \end{description} \begin{theorem} \label{3}Nech $\mathbf{a}$ je hromadn\'{y} bod mno\v{z}iny $A\subset \mathbf{% R}^{n}$ a nech $\mathbf{f}:A\left( \subset \mathbf{R}^{n}\right) \longrightarrow \mathbf{R}^{m}.$Potom $\lim_{\mathbf{x}\longrightarrow \mathbf{a}}\mathbf{f}\left( \mathbf{x}\right) =\mathbf{b\Longleftrightarrow } $ ak pre ka\v{z}d\'{u} postupnos\v{t} $\left\{ \mathbf{x}^{\left( k\right) }\right\} _{k=1}^{\infty }\subset A,$ tak\'{u} \v{z}e $\lim_{k% \longrightarrow \infty }\mathbf{x}^{\left( k\right) }=\mathbf{a},\,\mathbf{x}% ^{\left( k\right) }\neq \mathbf{a},\,\forall k\in \mathbf{N}$ plat\'{\i} $% \lim_{k\longrightarrow \infty }\mathbf{f}\left( \mathbf{x}^{\left( k\right) }\right) =\mathbf{b.}$ \end{theorem} \begin{tabular}{|c|} \hline {\small %TCIMACRO{\hyperref{D\^{o}kaz}{}{}{DO222.tex}}% %BeginExpansion \msihyperref{D\^{o}kaz}{}{}{DO222.tex}% %EndExpansion } \\ \hline \end{tabular}% \label{2} Pod\v{l}a predch\'{a}dzaj\'{u}cej vety a %TCIMACRO{\hyperref{vety}{}{}{Ma13.tex#12} }% %BeginExpansion \msihyperref{vety}{}{}{Ma13.tex#12} %EndExpansion sa \'{u}vahy o limite funkcie $\mathbf{f}:A\left( \subset \mathbf{R}% ^{n}\right) \longrightarrow \mathbf{R}^{m}$ daj\'{u} redukova\v{t} na $m$ pr% \'{\i}padov lim\'{\i}t funkci\'{\i} $f_{i}:A\left( \subset \mathbf{R}% ^{n}\right) \longrightarrow \mathbf{R},\,i=1,2,...,m.$ \begin{theorem} \label{13}Nech $\mathbf{f}:A\left( \subset \mathbf{R}^{n}\right) \longrightarrow \mathbf{R}^{m},\,$\ ($\mathbf{a}$ je hromadn\'{y} bod mno% \v{z}iny $A)$ m\'{a} komponenty $\mathbf{f}=(f_{1},f_{2},...,f_{m}).$ Potom $% \lim_{\mathbf{x}\longrightarrow \mathbf{a}}\mathbf{f}\left( \mathbf{x}% \right) =\mathbf{b\Longleftrightarrow }$ ak $\lim_{\mathbf{x}\longrightarrow \mathbf{a}}f_{i}\left( \mathbf{x}\right) =b_{i},\,\forall i=1,2,...,m.$ \end{theorem} \begin{tabular}{|c|} \hline {\small %TCIMACRO{\hyperref{D\^{o}kaz}{}{}{DO223.tex}}% %BeginExpansion \msihyperref{D\^{o}kaz}{}{}{DO223.tex}% %EndExpansion } \\ \hline \end{tabular}% \label{4} \begin{description} \item[Pozn\'{a}mka] Tento v\'{y}sledok znamen\'{a}, \v{z}e preto aby sme na% \v{s}li limitu funkcie $\mathbf{f}:A\left( \subset \mathbf{R}^{n}\right) \longrightarrow \mathbf{R}^{m}$\ je nutn\'{e} a sta\v{c}\'{\i}, aby sme vypo% \v{c}\'{\i}tali limity funkci\'{\i} $f_{i}:A\left( \subset \mathbf{R}% ^{n}\right) \longrightarrow \mathbf{R,\,}i=1,2,\dots ,m.$ \end{description} \begin{theorem} \label{14}Nech pre $\mathbf{f,g}:A\left( \subset \mathbf{R}^{n}\right) \longrightarrow \mathbf{R}^{m}$ je $\lim_{\mathbf{x}\longrightarrow \mathbf{a% }}\mathbf{f}\left( \mathbf{x}\right) =\mathbf{b},\,\lim_{\mathbf{x}% \longrightarrow \mathbf{a}}\mathbf{g}\left( \mathbf{x}\right) =\mathbf{c}$ a nech $\alpha \in \mathbf{R}$ ($\mathbf{a}$ je hromadn\'{y} bod mno\v{z}iny $% A $). Potom a) $\lim_{\mathbf{x}\longrightarrow \mathbf{a}}\left( \mathbf{f+g}\right) \left( \mathbf{x}\right) =\lim_{\mathbf{x}\longrightarrow \mathbf{a}}\mathbf{% f}\left( \mathbf{x}\right) +\lim_{\mathbf{x}\longrightarrow \mathbf{a}}% \mathbf{g}\left( \mathbf{x}\right) =\mathbf{b+c,}$ b) $\lim_{\mathbf{x}\longrightarrow \mathbf{a}}\left( \alpha \mathbf{f}% \right) \left( \mathbf{x}\right) =\lim_{\mathbf{x}\longrightarrow \mathbf{a}% }\alpha \mathbf{f}\left( \mathbf{x}\right) =\alpha \mathbf{b,}$ c) ak $m=1,$ potom $\lim_{\mathbf{x}\longrightarrow \mathbf{a}}\left( fg\right) \left( \mathbf{x}\right) =\left( \lim_{\mathbf{x}\longrightarrow \mathbf{a}}f\left( \mathbf{x}\right) \right) \left( \lim_{\mathbf{x}% \longrightarrow \mathbf{a}}g\left( \mathbf{x}\right) \right) =bc,$ d) ak $m=1$ a $c\neq 0,$ potom $\lim_{\mathbf{x}\longrightarrow \mathbf{a}% }\left( \frac{1}{g}\right) \left( \mathbf{x}\right) =\lim_{\mathbf{x}% \longrightarrow \mathbf{a}}\frac{1}{g\left( \mathbf{x}\right) }=\frac{1}{c}.$ \end{theorem} \begin{tabular}{|c|} \hline {\small %TCIMACRO{\hyperref{D\^{o}kaz}{}{}{DO224.tex}}% %BeginExpansion \msihyperref{D\^{o}kaz}{}{}{DO224.tex}% %EndExpansion } \\ \hline \end{tabular}% \label{5} \begin{description} \item[Pozn\'{a}mka] Niekedy k priestoru $\mathbf{R}^{n},\,n\geq 2$\ prid\'{a}% vame jeden \quotedblbase bod`` \ $\infty $\ \ a priestor $\overline{\mathbf{R% }^{n}}=\mathbf{R}^{n}\cup \left\{ \infty \right\} $ naz\'{y}vame jednobodov% \'{e} roz\v{s}\'{\i}renie priestoru $\mathbf{R}^{n}.$\ Ak definujeme okolie a prstencov\'{e} okolie bodu $\infty $\ nasleduj\'{u}cim sp\^{o}sobom: \ $% \varepsilon $-\emph{okol\'{\i}m bodu} $\infty $ naz\'{y}vame mno\v{z}inu \[ O_{\varepsilon }\left( \infty \right) =\left\{ \mathbf{x}\in \mathbf{R}% ^{n};\,\left\| \mathbf{x}\right\| >\frac{1}{\varepsilon }\right\} , \] a prstencov\'{y}m $\varepsilon $-\emph{okol\'{\i}m bodu} $\infty $ naz\'{y}% vame mno\v{z}inu \[ O_{\varepsilon }^{\circ }\left( \infty \right) =O_{\varepsilon }\left( \infty \right) , \] potom tvrdenia %TCIMACRO{\hyperref{vety}{}{}{Ma22.tex#14} }% %BeginExpansion \msihyperref{vety}{}{}{Ma22.tex#14} %EndExpansion zost\'{a}vaj\'{u} v platnosti aj pre nevlastn\'{e} limity \end{description} \begin{example} Nech $f:A\left( \subset \mathbf{R}^{n}\right) \longrightarrow \mathbf{R}% ,\,f\left( x,y\right) =xy.$ Potom ak $\mathbf{x}=(x,y)$ a $\mathbf{a}=(a,b)$ tak $\lim_{\left( x,y\right) \longrightarrow \left( a,b\right) }f\left( x,y\right) =ab.$ \end{example} \begin{solution} $f(x,y)=\pi _{1}(x,y)\pi _{2}(x,y)=xy.$ Pod\v{l}a predch\'{a}dzaj\'{u}cej vety - \v{c}asti c) dost\'{a}vame \[ \lim_{\mathbf{x}\longrightarrow \mathbf{a}}f\left( \mathbf{x}\right) =\lim_{\left( x,y\right) \longrightarrow \left( a,b\right) }xy=\lim_{\left( x,y\right) \longrightarrow \left( a,b\right) }\pi _{1}(x,y)\pi _{2}(x,y)=ab.\square \] \end{solution} \begin{example} Nech $f:\mathbf{R}^{2}\setminus \left\{ \left( 0,0\right) \right\} \longrightarrow \mathbf{R},\,f\left( x,y\right) =\frac{x^{3}+y^{3}}{% x^{2}+y^{2}}.$ Vypo\v{c}\'{\i}tajte $\lim_{\left( x,y\right) \longrightarrow \left( -1,2\right) }f\left( x,y\right) .$ \end{example} \begin{solution} Pod\v{l}a predch\'{a}dzaj\'{u}cej vety - \v{c}asti c), d) plat\'{\i} \[ \lim_{\left( x,y\right) \longrightarrow \left( -1,2\right) }f\left( x,y\right) =\lim_{\left( x,y\right) \longrightarrow \left( -1,2\right) }% \frac{x^{3}+y^{3}}{x^{2}+y^{2}}=\frac{\lim_{\left( x,y\right) \longrightarrow \left( -1,2\right) }\left( x^{3}+y^{3}\right) }{\lim_{\left( x,y\right) \longrightarrow \left( -1,2\right) }\left( x^{2}+y^{2}\right) }=% \frac{\left( -1\right) ^{3}+2^{3}}{\left( -1\right) ^{2}+2^{2}}=\frac{7}{5}% .\square \] \end{solution} Existuje e\v{s}te jeden sp\^{o}sob kombin\'{a}cie funkci\'{\i} a to zlo\v{z}% en\'{a} funkcia. Ak $\mathbf{f}:A\left( \subset \mathbf{R}^{n}\right) \longrightarrow \mathbf{% R}^{p},\,\mathbf{g}:B\left( \subset \mathbf{R}^{p}\right) \longrightarrow \mathbf{R}^{m}$ a ak $\mathbf{f}(A)\subset B,$ potom \label{12}\emph{zlo\v{z}% en\'{a} funkcia (zobrazenie)} $\mathbf{g\circ f}:A\left( \subset \mathbf{R}% ^{n}\right) \longrightarrow \mathbf{R}^{m}$ je definovan\'{a} vz\v{t}ahom $(% \mathbf{g\circ f})(\mathbf{x})=\mathbf{g}(\mathbf{f}(\mathbf{x})),$ pre $% \mathbf{x}\in A.$ \begin{description} \item[Pozn\'{a}mka] Skladanie funkci\'{\i} \textsl{nie je} komutat\'{\i}vne. \end{description} \begin{theorem} Nech $\mathbf{f}:A\left( \subset \mathbf{R}^{n}\right) \longrightarrow \mathbf{R}^{m}$ a $\mathbf{g}:B\left( \subset \mathbf{R}^{p}\right) \longrightarrow \mathbf{R}^{m}$ pri\v{c}om $\mathbf{f}(A)\subset B.$ Nech $% \lim_{\mathbf{x}\longrightarrow \mathbf{a}}\mathbf{f}\left( \mathbf{x}% \right) =\mathbf{c}$ a v nejakom $O_{\delta }^{\circ }\left( \mathbf{a}% \right) $ plat\'{\i}, \v{z}e $\mathbf{f}(\mathbf{x})\neq \mathbf{c}$ ($% \mathbf{a}$ je hromadn\'{y} bod $A,$ $\mathbf{c}$ je hromadn\'{y} bod $B$) . Ak $\lim_{\mathbf{y}\longrightarrow \mathbf{c}}\mathbf{g}\left( \mathbf{y}% \right) =\mathbf{b,}$ potom $\lim_{\mathbf{x}\longrightarrow \mathbf{a}% }\left( \mathbf{g\circ f}\right) \left( \mathbf{x}\right) =\lim_{\mathbf{x}% \longrightarrow \mathbf{a}}\mathbf{g}\left( \mathbf{f}\left( \mathbf{x}% \right) \right) =\mathbf{b.}$ \end{theorem} \begin{tabular}{|c|} \hline {\small %TCIMACRO{\hyperref{D\^{o}kaz}{}{}{DO225.tex}}% %BeginExpansion \msihyperref{D\^{o}kaz}{}{}{DO225.tex}% %EndExpansion } \\ \hline \end{tabular}% \label{6} \begin{example} Nech $f:A\longrightarrow \mathbf{R},\,f\left( x,y\right) =\ln \frac{x}{y}.$ N% \'{a}jdite defini\v{c}n\'{y} obor funkcie $f$\ a vypo\v{c}\'{\i}tajte $% \lim_{\left( x,y\right) \longrightarrow \left( e,1\right) }f\left( x,y\right) .$ \end{example} \begin{solution} M\'{a}me \[ A=D(f)=\{(x,y)\in \mathbf{R}^{2};((x>0)\wedge (y>0))\vee ((x<0)\wedge (y<0))\}, \]% \[ \lim_{\left( x,y\right) \longrightarrow \left( e,1\right) }f\left( x,y\right) =\lim_{\left( x,y\right) \longrightarrow \left( e,1\right) }\ln \frac{x}{y}=\ln \frac{e}{1}=1.\square \] \end{solution} \begin{description} \item[Pozn\'{a}mka] Pre $f:A\left( \subset \mathbf{R}^{n}\right) \longrightarrow \mathbf{R}$ s\'{u} platn\'{e} v\v{s}etky vety o limit\'{a}ch funkcie jednej re\'{a}lnej premennej. Ako pr\'{\i}klad uvedieme bez d\^{o}% kazu aspo\v{n} najd\^{o}le\v{z}itej\v{s}ie z nich: vetu o nerovnosti medzi limitami a vetu o nevlastn\'{y}ch limit\'{a}ch. \end{description} \begin{theorem} Nech $f,g,h:A\left( \subset \mathbf{R}^{n}\right) \longrightarrow \mathbf{R}% . $Nech \[ f(\mathbf{x})\leq g(\mathbf{x})\leq h(\mathbf{x}),\,\forall \mathbf{x}\in A. \] Ak $\lim_{\mathbf{x}\longrightarrow \mathbf{a}}f\left( \mathbf{x}\right) =\lim_{\mathbf{x}\longrightarrow \mathbf{a}}h\left( \mathbf{x}\right) =L,$ potom aj $\lim_{\mathbf{x}\longrightarrow \mathbf{a}}g\left( \mathbf{x}% \right) $ existuje a $\lim_{\mathbf{x}\longrightarrow \mathbf{a}}g\left( \mathbf{x}\right) =L.$ \end{theorem} \begin{example} Nech $f:\mathbf{R}^{2}\setminus \left\{ \left( 0,0\right) \right\} \longrightarrow \mathbf{R},\,f\left( x,y\right) =\frac{x^{3}+y^{3}}{% x^{2}+y^{2}}.$ Vypo\v{c}\'{\i}tajte $\lim_{\left( x,y\right) \longrightarrow \left( 0,0\right) }f\left( x,y\right) .$ \end{example} \begin{solution} Plat\'{\i} \[ 0\leq \frac{x^{2}}{x^{2}+y^{2}}\leq 1,\,0\leq \frac{y^{2}}{x^{2}+y^{2}}\ \ \leq 1, \]% odkia\v{l} pod\v{l}a vety o nerovnostiach medzi limitami m\'{a}me: \[ \lim_{\left( x,y\right) \longrightarrow \left( 0,0\right) }f\left( x,y\right) =\lim_{\left( x,y\right) \longrightarrow \left( 0,0\right) }\frac{% x^{3}+y^{3}}{x^{2}+y^{2}}=\lim_{\left( x,y\right) \longrightarrow \left( 0,0\right) }\left[ x\frac{x^{2}}{x^{2}+y^{2}}+y\frac{y^{2}}{x^{2}+y^{2}}% \right] =0.\square \] \end{solution} \begin{theorem} Nech $f,g,h:A\left( \subset \mathbf{R}^{n}\right) \longrightarrow \mathbf{R}% . $ a) Ak existuje $\lim_{\mathbf{x}\longrightarrow \mathbf{a}}f\left( \mathbf{x}% \right) ,$ tak $\lim_{\mathbf{x}\longrightarrow \mathbf{a}}\left( -f\right) \left( \mathbf{x}\right) =-\lim_{\mathbf{x}\longrightarrow \mathbf{a}% }f\left( \mathbf{x}\right) ,$ b) Ak $\lim_{\mathbf{x}\longrightarrow \mathbf{a}}f\left( \mathbf{x}\right) =\infty $ a $\forall \mathbf{x}\in A$ je $g(\mathbf{x})\geq k,$ tak $\lim_{% \mathbf{x}\longrightarrow \mathbf{a}}\left( f+g\right) \left( \mathbf{x}% \right) =\infty ,$ c) Ak $\lim_{\mathbf{x}\longrightarrow \mathbf{a}}f\left( \mathbf{x}\right) =\infty $ a a $\forall \mathbf{x}\in A$ je $g(\mathbf{x}):>k>0,$ tak $\lim_{% \mathbf{x}\longrightarrow \mathbf{a}}\left( fg\right) \left( \mathbf{x}% \right) =\infty ,$ d) Ak $\lim_{\mathbf{x}\longrightarrow \mathbf{a}}\left| f\right| \left( \mathbf{x}\right) =\infty $ a $\forall \mathbf{x}\in A$ je $f(\mathbf{x}% )\neq 0,$ tak $\lim_{\mathbf{x}\longrightarrow \mathbf{a}}\left( \frac{1}{f}% \right) \left( \mathbf{x}\right) =0,$ e) Ak $\lim_{\mathbf{x}\longrightarrow \mathbf{a}}f\left( \mathbf{x}\right) =0$ a $\forall \mathbf{x}\in A$ je $f(\mathbf{x})>0,$ tak $\lim_{\mathbf{x}% \longrightarrow \mathbf{a}}\left( \frac{1}{f}\right) \left( \mathbf{x}% \right) =\infty .$ \end{theorem} \begin{example} Nech $f:\mathbf{R}^{2}\setminus \left\{ \left( 0,0\right) \right\} \longrightarrow \mathbf{R},\,f\left( x,y\right) =\frac{1}{x^{2}+y^{2}}.$ Vypo% \v{c}\'{\i}tajte $\lim_{\left( x,y\right) \longrightarrow \left( 0,0\right) }f\left( x,y\right) .$ \end{example} \begin{solution} Plat\'{\i} \[ x^{2}+y^{2}>0,\,\forall \mathbf{x}\in \mathbf{R}^{2}\setminus \left\{ \left( 0,0\right) \right\} \text{ \ a \ }\lim_{\left( x,y\right) \longrightarrow \left( 0,0\right) }x^{2}+y^{2}=0, \]% potom pod\v{l}a \v{c}asti e) predch\'{a}dzaj\'{u}cej vety: \[ \lim_{\left( x,y\right) \longrightarrow \left( 0,0\right) }f\left( x,y\right) =\lim_{\left( x,y\right) \longrightarrow \left( 0,0\right) }\frac{% 1}{x^{2}+y^{2}}=\infty .\square \] \end{solution} \begin{theorem} Nech $\mathbf{f}:A\left( \subset \mathbf{R}^{n}\right) \longrightarrow \mathbf{R}^{m}$ a $\mathbf{a\in \,}int\left( A\right) .$ Ak $\lim_{\mathbf{x}% \longrightarrow \mathbf{a}}\mathbf{f}\left( \mathbf{x}\right) =\mathbf{b,}$ tak $\lim_{t\longrightarrow 0}\mathbf{f}\left( \mathbf{a}+t\mathbf{u}\right) =\mathbf{b}$ pre ka\v{z}d\'{y} smer $\mathbf{u.}$ \end{theorem} \begin{tabular}{|c|} \hline {\small %TCIMACRO{\hyperref{D\^{o}kaz}{}{}{DO226.tex}}% %BeginExpansion \msihyperref{D\^{o}kaz}{}{}{DO226.tex}% %EndExpansion } \\ \hline \end{tabular}% \label{7} \begin{description} \item[Pozn\'{a}mka] Opa\v{c}n\'{e} tvrdenie neplat\'{\i}. \end{description} \begin{example} Nech $f:\mathbf{R}^{2}\setminus \left\{ \left( 0,0\right) \right\} \longrightarrow \mathbf{R},\,f\left( x,y\right) =\frac{xy^{2}}{x^{2}+y^{4}}.$ Uk\'{a}\v{z}te, \v{z}e $\lim_{\left( x,y\right) \longrightarrow \left( 0,0\right) }f\left( x,y\right) $ neexistuje. \end{example} \begin{solution} Pozd\'{l}\v{z} ka\v{z}dej priamky $M=\left\{ \mathbf{x}(t)=\mathbf{0}+t% \mathbf{u}=\left( 0,0\right) +t\left( u,v\right) =(tu,tv),\,t\in \mathbf{R}% \right\} ,$ kde $u\neq 0$ alebo $v\neq 0$ m\'{a}me \[ \lim_{t\longrightarrow 0}\frac{t^{3}uv^{2}}{t^{2}u^{2}+t^{4}u^{4}}% =\lim_{t\longrightarrow 0}t\frac{uv^{2}}{u^{2}+t^{2}u^{4}}=0, \]% ale ak sa bl\'{\i}\v{z}ime k bodu $\mathbf{a}=(0,0)$ napr\'{\i}klad po krivke $L=\left\{ \mathbf{x}(t)=\{(t,\sqrt{t}),\,t\in \left\langle 0,\infty \right) \right\} ,$ m\'{a}me \[ \lim_{\left( x,y\right) \longrightarrow \left( 0,0\right) }\frac{xy^{2}}{% x^{2}+y^{4}}=\lim_{\left( t,\sqrt{t}\right) \longrightarrow \left( 0,0\right) }\left[ \frac{t^{2}}{t^{2}+t^{2}}\right] =\frac{1}{2}, \]% \v{c}o znamen\'{a}, \v{z}e $\lim_{\left( x,y\right) \longrightarrow \left( 0,0\right) }\frac{xy^{2}}{x^{2}+y^{4}}$ neexistuje. $\square $ \end{solution} \begin{description} \item[Pozn\'{a}mka] Z %TCIMACRO{\hyperref{pozn\'{a}mky}{}{}{Ma22.tex#10} }% %BeginExpansion \msihyperref{pozn\'{a}mky}{}{}{Ma22.tex#10} %EndExpansion plynie, \v{z}e ak pre dve podmno\v{z}iny $L,\,M\subset A$ defini\v{c}n\'{e}% ho oboru funkcie $\mathbf{f},$ ktor\'{y}ch hromadn\'{y}m bodom je bod $% \mathbf{a},$ plat\'{\i} $\lim_{\mathbf{x}\longrightarrow \mathbf{a,x}\in L}% \mathbf{f}\left( \mathbf{x}\right) \neq \lim_{\mathbf{x}\longrightarrow \mathbf{a,x}\in M}\mathbf{f}\left( \mathbf{x}\right) ,$ potom $\lim_{\mathbf{% x}\longrightarrow \mathbf{a}}\mathbf{f}\left( \mathbf{x}\right) $ neexistuje. \end{description} \begin{center} \begin{tabular}{|c|c|c|c|c|c|c|} \hline \textbf{% %TCIMACRO{\hyperref{Obsah}{}{}{maiindex.tex}}% %BeginExpansion \msihyperref{Obsah}{}{}{maiindex.tex}% %EndExpansion } & \textbf{% %TCIMACRO{\hyperref{Obsah kapitoly}{}{}{Ma2.tex}}% %BeginExpansion \msihyperref{Obsah kapitoly}{}{}{Ma2.tex}% %EndExpansion } & \textbf{% %TCIMACRO{\hyperref{Predch\'{a}dzaj\'{u}ca str\'{a}nka}{}{}{Ma21.tex}}% %BeginExpansion \msihyperref{Predch\'{a}dzaj\'{u}ca str\'{a}nka}{}{}{Ma21.tex}% %EndExpansion } & \textbf{% %TCIMACRO{\hyperref{Nasleduj\'{u}ca str\'{a}nka}{}{}{Ma23.tex}}% %BeginExpansion \msihyperref{Nasleduj\'{u}ca str\'{a}nka}{}{}{Ma23.tex}% %EndExpansion } & \textbf{% %TCIMACRO{\hyperref{Ot\'{a}zky}{}{}{O2.tex}}% %BeginExpansion \msihyperref{Ot\'{a}zky}{}{}{O2.tex}% %EndExpansion } & \textbf{% %TCIMACRO{\hyperref{Cvi\v{c}enia}{}{}{C2.tex}}% %BeginExpansion \msihyperref{Cvi\v{c}enia}{}{}{C2.tex}% %EndExpansion } & \textbf{% %TCIMACRO{\hyperref{Index}{}{}{Glos.tex}}% %BeginExpansion \msihyperref{Index}{}{}{Glos.tex}% %EndExpansion } \\ \hline \end{tabular} \end{center} \rule{6.5in}{0.04in} \textsl{Matematick\'{a} anal\'{y}za II} \section{Funkcie, limita funkcie, spojit\'{e} funkcie} \end{document}