\documentclass{article} \usepackage{amssymb} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %TCIDATA{OutputFilter=LATEX.DLL} %TCIDATA{Created=Monday, June 25, 2001 17:45:57} %TCIDATA{LastRevised=Saturday, June 01, 2002 19:49:33} %TCIDATA{} %TCIDATA{} %TCIDATA{CSTFile=On line bluem.cst} %TCIDATA{PageSetup=72,72,72,72,0} %TCIDATA{AllPages= %H=36 %F=36,\PARA{038
\QTR{small}{Matematick\U{e1} anal\U{fd}za I online - D\U{f4}kazy\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{Aplik\'{a}cie deriv\'{a}ci\'{\i}} \subsection{D\^{o}kaz L' Hospitalovho pravidla} \textbf{D\^{o}kaz: }Nech $\varepsilon >0$ je \v{l}ubovo\v{l}n\'{e}. Potom pre $O_{\frac{\varepsilon }{2}}\left( L\right) $ existuje $O_{\delta }^{\circ }\left( b\right) $ tak\'{e}, \v{z}e $\frac{f\,\/\,^{\prime }}{% g^{\prime }}\left( O_{\delta }^{\circ }\left( b\right) \cap \left\langle a,b\right) \right) \subset O_{\frac{\varepsilon }{2}}\left( L\right) .$ Nech $x,y\in O_{\delta }^{\circ }\left( b\right) \cap \left\langle a,b\right) ,\,x\neq y.$ Na intervale $\left\langle x,y\right\rangle $, resp $% \left\langle y,x\right\rangle $ s\'{u} splnen\'{e} predpoklady Cauchyho vety. Preto \[ \frac{f\left( x\right) -f\left( y\right) }{g\left( x\right) -g\left( y\right) }=\frac{f\,\/\,^{\prime }\left( c\left( x\right) \right) }{% g^{\prime }\left( c\left( x\right) \right) }, \]% kde $c(x)\in (x,y)$ resp. $c(x)\in (y,x)$. Potom m\'{a}me $c(x)\in O_{\delta }^{\circ }\left( b\right) \cap \left\langle a,b\right) $. Teda \[ \frac{f\,\/\,^{\prime }\left( c\left( x\right) \right) }{g^{\prime }\left( c\left( x\right) \right) }=\frac{f\left( x\right) -f\left( y\right) }{% g\left( x\right) -g\left( y\right) }\in O_{\frac{\varepsilon }{2}}\left( L\right) . \]% Z toho vypl\'{y}va, \v{z}e $\lim_{y\longrightarrow b}\frac{f\left( x\right) -f\left( y\right) }{g\left( x\right) -g\left( y\right) }=\frac{f\left( x\right) }{g\left( x\right) }\in O_{\varepsilon }\left( L\right) ,$ teda $% \left( \frac{f}{g}\right) \left( O_{\delta }^{\circ }\left( b\right) \cap \left\langle a,b\right) \right) \subset O_{\varepsilon }\left( L\right) ,$% t.j. \[ \lim_{x\longrightarrow b}\frac{f\left( x\right) }{g\left( x\right) }% =L.\blacksquare \] \begin{center} \begin{tabular}{|c|} \hline \hyperref{{\small Sp\"{a}\v{t}}}{}{}{M64.tex#1} \\ \hline \end{tabular} \end{center} \end{document}