Verze z 15. 10. 2014, 20:25

Zdrojový kód

%\wikiskriptum{Matematika1Priklady}
\section{Limity a spojitost}
 
\subsection*{\fbox{Rozcvička}}
V této krátké části jsou příklady, které pro svou nižší náročnost nebudou ve zkouškové písemce, a tudíž nejsou číslovány.
 
\begin{itemize}
 
%\input{01}
 
\item \begin{priklad}
\lim_{x \to +\infty}\frac{\sqrt{x^2+1}}{x+1}
\end{priklad}
\res{1}
 
\item \begin{priklad}
\lim_{x \to \pi} \frac{\sin{x}}{x-\pi}
\end{priklad}
\res{-1}
 
\item \begin{priklad}
\lim_{x \to 0} x \cot{3x} 
\end{priklad}
\res{$\frac{1}{3}$}
 
\item \begin{priklad}
\lim_{x \to 1} \frac{\ln{x}}{x^2-1} 
\end{priklad}
\res{$\frac{1}{2}$}
 
\item \begin{priklad}
\lim_{x \to 0} \frac{x^3+2x^2-x-2}{x^2-1}
\end{priklad}
\res{2}
 
\item \begin{priklad}
\lim_{x \to +\infty} \frac{3x^2 - 2x + 5}{ 4x^2 + 3x -7}
\end{priklad}
\res{$\frac{3}{4}$}
 
 
\item \begin{priklad}
\lim_{x \to 3} \frac{x^2-9}{x^2+2x-15}
\end{priklad}
\res{$\frac{3}{4}$}
 
\item \begin{priklad}
\lim_{x \to 0} \frac{\tan{x}}{x} 
\end{priklad}
\res{1}
 
\item \begin{priklad}
\lim_{x \to 0} x \cot{x} 
\end{priklad}
\res{1}
 
\item \begin{priklad}
\lim_{x \to 0} \frac{\sin{5x}}{x} 
\end{priklad}
\res{5}
 
\item \begin{priklad}
\lim_{x \to 5} \frac{x^2-4x-5}{x^2-7x+10}
\end{priklad}
\res{2}
 
\item \begin{priklad}
\lim_{x \to +\infty} \frac{5x^3 - x^2 +3}{x^3+6x^2-4} 
\end{priklad}
\res{5}
 
\item \begin{priklad}
\lim_{x \to 1} \frac{x^2+x-2}{4x-4}
\end{priklad}
\res{$\frac{3}{4}$}
 
\item \begin{priklad}
\lim_{x \to 0} \frac{\cos^2{x} - 1}{x^2}
\end{priklad}
\res{-1}
 
 
\item \begin{priklad}
\lim_{x \to +\infty} \frac{x^3-1}{5x^3}
\end{priklad}
\res{$\frac{1}{5}$}
 
\item \begin{priklad}
\lim_{x \to 2} \frac{3x^3-10x-4}{4-x^2} 
\end{priklad}
\res{$- \frac{13}{2}$}
 
\item \begin{priklad}
\lim_{x \to -1} \frac{2x^2-6x-8}{2x^3+2} 
\end{priklad}
\res{$- \frac{5}{3}$}
 
\item \begin{priklad}
\lim_{x \to a} \frac{\sin{x} - \sin{a}}{x - a} 
\end{priklad}
\res{$\cos{a}$}
 
\end{itemize}
 
\subsection*{\fbox{Zkouškové příklady}}
 
 
\begin{enumerate}
 
\odstavec{l'Hôpitalovo pravidlo zakázáno}
 
\item  \bezlh
\begin{priklad}
\lim_{x \to 3} \frac{x^2+x-12}{9-3x} 
\end{priklad}
\res{$ - \frac{7}{3}$}
 
\item  \bezlh
\begin{priklad}
\lim_{x \to 0} \frac{\cos{x} - 1}{x^2} 
\end{priklad}
\res{$- \frac{1}{2}$}
 
 
\item  \bezlh
\begin{priklad}
\lim_{x \to 2} \frac{x^3-8}{x^4-16} 
\end{priklad}
\res{$\frac{3}{8}$}
 
\item \bezlh
\begin{priklad}
\lim_{x \to 2} \frac{x^2 - 2x}{8-x^3} 
\end{priklad}
\res{$- \frac{1}{6}$}
 
 
\item \bezlh 
\begin{priklad}
\lim_{x \to 0} \frac{\sin^2 x + \sin^8 x - \sin^3 x}{\sin^5 x + 1 - \cos^2 x + \sin^7 x}
\end{priklad}
\res{1}
 
 
\item \bezlh
\begin{priklad}
\lim_{x \to 0} \frac{\sqrt{\sin^2 x}+\sqrt{1-\cos^2 x}}{|x|}
\end{priklad}
\res{2}
 
\item \bezlh
\begin{priklad}
\lim_{x \to -2} \frac{x^3 + 3x^2+3x+2}{x^3+3x^2+4x+4}
\end{priklad}
\res{$\frac34$}
 
\item \bezlh
\begin{priklad}
\lim_{x \to 0} \frac{x^3+2x^2+5x}{(x^2+6)\sin x} 
\end{priklad}
\res{$\frac56$}
 
\item \bezlh
\begin{priklad}
\lim_{x \to 0} \frac{\sqrt{1 + \tg x} - \sqrt{1+\sin x}}{x^3}
\end{priklad}
\res{$\frac14$}
 
\item \bezlh
\begin{priklad}
\lim_{x \to 0} \frac{\sqrt{1+\sin x}-\sqrt{1-\sin x}}{x}
\end{priklad}
\res{1}
 
\item \bezlh
\begin{priklad}
\lim_{x \to 5} \frac{x^2 + 2x-35}{x^3-3x^2-9x-5}
\end{priklad}
\res{$\frac13$}
 
\item \bezlh
\begin{priklad}
\lim_{x \to 0} \frac{\sin^2x}{x\sqrt{1-\cos^2x}}
\end{priklad}
\res{neex}
 
\item \bezlh
\begin{priklad}
\lim_{x \to -2} \frac{x^3+2x^2+x+2}{x^2+3x+2}
\end{priklad}
\res{-5}
 
\item \bezlh
\begin{priklad}
\lim_{x \to +\infty} \frac{x^3}{2x^2-1} - \frac{x^2}{2x+1}
\end{priklad}
\res{$\frac14$}
 
\item \bezlh
\begin{priklad}
\lim_{x \to 3} \frac{\sqrt{x+13} - 2 \sqrt{x+1}}{x^2-9}
\end{priklad}
\res{$-\frac{1}{16}$}
 
\item \bezlh
\begin{priklad}
\lim_{x \to 0} x~\hbox{tg}{\left ( \frac{\pi}{2}-x \right )}
\end{priklad}
\res{1}
 
\item \bezlh
\begin{priklad}
\lim_{x \to 0} \frac{x^2}{\sqrt{1+x \sin x} - \sqrt{\cos x}}
\end{priklad}
\res{$\frac43$}
 
\item \bezlh
\begin{priklad}
\lim_{x \to 2} \frac{x^3-4x}{x^4+x-18}
\end{priklad}
\res{$\frac{8}{33}$}
 
\item \bezlh
\begin{priklad}
\lim_{x \to 2} \frac{1-\cos(x-2)}{x^3-2x^2-4x+8}  
\end{priklad}
\res{$\frac18$}
 
\item \bezlh
\begin{priklad}
\lim_{x \to +\infty} \frac{\sinh x}{\cosh x + \sinh x} 
\end{priklad}
\res{$\frac12$}
 
\item \bezlh
\begin{priklad}
\lim_{x \to -\infty} \frac{\sinh x}{\cosh x + \sinh x} 
\end{priklad}
\res{$-\infty$}
 
\item \bezlh
\begin{priklad}
\lim_{x \to 0} \frac{1-\sqrt{\cos x}}{x^2}
\end{priklad}
\res{$\frac14$}
 
\item \bezlh
\begin{priklad}
\lim_{x \to 0} \frac{\cos^2 x + \sin^8 x - \cos^3 x}{\sin^5 x + 1 - \cos^2 x + \sin^7 x}
\end{priklad}
\res{$\frac12$}
 
\item \bezlh
\begin{priklad}
\lim_{x \to 4} \frac{\sqrt{1+2x}-3}{\sqrt{x} - 2}
\end{priklad}
\res{$\frac{4}{3}$}
 
\item \bezlh
\begin{priklad}
\lim_{x \to -\infty}\frac{\sqrt{x^2+1}}{x+1}
\end{priklad}
\res{-1}
 
\item \bezlh
\begin{priklad}
\lim_{x \to 0} \frac{x^2}{\sqrt{1+x \sin{x}} - \sqrt{\cos{x}}}
\end{priklad}
\res{$\frac{4}{3}$}
 
\item \bezlh
\begin{priklad}
\lim_{x \to 0} \frac{\tg{x} - \sin{x}}{\sin^3{x}}
\end{priklad}
\res{$\frac{1}{2}$}
 
\item \bezlh
\begin{priklad}
\lim_{x \to 4} \big( \frac{1}{x} - \frac{1}{4} \big) \big(\frac{1}{x-4} \big)
\end{priklad}
\res{$ -\frac{1}{16}$}
 
\item \bezlh
\begin{priklad}
\lim_{x \to 0} \frac{x^2-3x}{\tg{x}} 
\end{priklad}
\res{-3}
 
\item \bezlh
\begin{priklad}
\lim_{x \to 1} (1-x) \tg{(x \frac{\pi}{2})} 
\end{priklad}
\res{$\frac{2}{\pi}$}
 
\item \bezlh
\begin{priklad}
\lim_{x \to \frac{\pi}{3}} \frac{\sin{(x-\frac{\pi}{3})}}{1-2\cos{x}}
\end{priklad}
\res{$\frac{\sqrt{3}}{3}$}
 
\item \bezlh
\begin{priklad}
\lim_{x \to \frac{\pi}{2}_-} \frac{\frac{\pi}{2} - x}{\sin{x} \cos{x}}
\end{priklad}
\res{1}
 
\item \bezlh
\begin{priklad}
\lim_{x \to \pi} \frac{\tg{x} - \sin{x}}{\cos ^3{x}} 
\end{priklad}
\res{0}
 
\item \bezlh
\begin{priklad}
\lim_{x \to 1} \frac{x^3+2x^2-x-2}{x^2-1}
\end{priklad}
\res{3}
 
\item \bezlh
\begin{priklad}
\lim_{x \to 0} \frac{1- \cos{2x} + \tg^2{x}}{x \sin{x}} 
\end{priklad}
\res{3}
 
\item \bezlh
\begin{priklad}
\lim_{x \to 1} \frac{x^2 - x}{\sqrt{x} - 1}
\end{priklad}
\res{2}
 
\item \bezlh
\begin{priklad}
\lim_{x \to +\infty} \frac{5-2x+ 3x^2}{3x-7+ 4x^2 } 
\end{priklad}
\res{$\frac{3}{4}$}
 
\item \bezlh
\begin{priklad}
\lim_{x \to +\infty} \Big(  \frac{x^3}{2x^2-1} - \frac{x^2}{2x+1}\Big)
\end{priklad}
\res{$\frac{1}{4}$}
 
\item \bezlh
\begin{priklad}
\lim_{x \to -2} \frac{\sqrt{6+x} - 2}{x+2} 
\end{priklad}
\res{$\frac{1}{4}$}
 
\item \bezlh
\begin{priklad}
\lim_{x \to 0} \frac{(1+x)^5 - (1+5x)}{x^2+x^5}
\end{priklad}
\res{10}
 
\item \bezlh
\begin{priklad}
\lim_{x \to 0} \frac{\sin{x} - x}{\sin{x} + x} 
\end{priklad}
\res{0}
 
\item \bezlh
\begin{priklad}
\lim_{x \to 2} \frac{x-2}{\sqrt{x+2} - 2}
\end{priklad}
\res{4}
 
\item \bezlh
\begin{priklad}
\lim_{x \to \frac{\pi}{4}} \frac{\cos{x} - \sin{x}}{1-\tg{x}}
\end{priklad}
\res{$\frac{\sqrt{2}}{2}$}
 
\item \bezlh
\begin{priklad}
\lim_{x \to +\infty} \frac{x^3 + 5x^4 - x^2 +3}{4-x^3+6x^2-x^4}
\end{priklad}
\res{-5}
 
\item  \bezlh
\begin{priklad}
\lim_{x \to 0} \frac{\sqrt{2+x} - \sqrt{2}}{\sin{x}}
\end{priklad}
\res{$\frac{\sqrt{2}}{4}$}
 
\item \bezlh
\begin{priklad}
\lim_{x \to 0} \frac{\cos{x} - 1}{1-\cos{x}^2}
\end{priklad}
\res{$-\frac{1}{2}$}
 
\item \bezlh
\begin{priklad}
\lim_{x \to 8} \frac{\sqrt{9+2x} - 5}{\sqrt[3]{x} - 2} 
\end{priklad}
\res{$\frac{12}{5}$}
 
\item \bezlh
\begin{priklad}
\lim_{x \to 0} \Big( \frac{\sin{3x}}{x} + \frac{\sin{x}}{3x}\Big)
\end{priklad}
\res{$\frac{10}{3}$}
 
\item \bezlh
\begin{priklad}
\lim_{x \to -1} \frac{x^2-2x-3}{x^2+x^3-2x-2} 
\end{priklad}
\res{4}
 
\item \bezlh
\begin{priklad}
\lim_{x \to 5} \frac{25 + x^2 -10x}{x^3  -9x- 3x^2-5}
\end{priklad}
\res{0}
 
\item \bezlh
\begin{priklad}
\lim_{x \to -1} \frac{x+1}{\sqrt{10+x} - 3}
\end{priklad}
\res{6}
 
\item \bezlh
\begin{priklad}
\lim_{x \to 0} \frac{1- \cos{x}}{\tg{x}}
\end{priklad}
\res{0}
 
\item \bezlh 
\begin{priklad}
\lim_{x \to -2} \frac{x^3+2x^2+x+2}{x^2+3x+2}
\end{priklad}
\res{-5}
 
\item \bezlh
\begin{priklad}
\lim_{x \to 2} \frac{x^3-4x}{x^4+x-18}
\end{priklad}
\res{$\frac{8}{33}$}
 
\item \bezlh
\begin{priklad}
\lim_{x \to 0} \frac{\sqrt{x+1} - 1}{\sin{x}}
\end{priklad}
\res{$\frac{1}{2}$}
 
\item \bezlh
\begin{priklad}
\lim_{x \to 1} \frac{\sqrt{x+3} - 2}{x - 1}
\end{priklad}
\res{$\frac{1}{4}$}
 
\item \bezlh
\begin{priklad}
\lim_{x \to 0} \frac{\sin{4x}}{\sqrt{x+1}-1} 
\end{priklad}
\res{8}
 
\item \bezlh  
\begin{priklad}
\lim_{x \to 0} \frac{1-\cos^2{x}}{x(1+\cos{x})}
\end{priklad}
\res{0}
 
\item \bezlh
\begin{priklad}
\lim_{x \to 3} \frac{9-x^2}{\sqrt{3x} - 3}
\end{priklad}
\res{-12}
 
\item \bezlh
\begin{priklad}
\lim_{x \to \frac{\pi}{4}} \frac{\sin{x} - \cos{x}}{\cos{2x}} 
\end{priklad}
\res{$-\frac{\sqrt{2}}{2}$}
 
\item \bezlh
\begin{priklad}
\lim_{x \to \pi} \frac{\sqrt{1-\tg{x}} - \sqrt{1+\tg{x}}}{\sin{2x}} 
\end{priklad}
\res{$- \frac{1}{2}$}
 
\item \bezlh
\begin{priklad}
\lim_{x \to a} \frac{\sin{x} - \sin{a}}{x^2 - a^2} 
\end{priklad}
\res{$\frac{\cos{a}}{2a}$}
 
\item \bezlh
\begin{priklad}
\lim_{x \to 0} \Big( \frac{1}{\sin{x}} - \frac{1}{\tg{x}}\Big)
\end{priklad}
\res{0}
 
\item \bezlh
\begin{priklad}
\lim_{x \to \frac{\pi}{2}} \Big( \frac{\sin{x}}{\cos^2{x}} - \tg^2{x} \Big)
\end{priklad}
\res{$\frac{1}{2}$}
 
\item \bezlh
\begin{priklad}
\lim_{x \to 1} \frac{\sqrt[3]{x} - 1}{\sqrt{x} - 1}
\end{priklad}
\res{$\frac{2}{3}$}
 
\item \bezlh
\begin{priklad}
\lim_{x \to 0} \frac{\sqrt[3]{1+x} - \sqrt[3]{1-x}}{x}
\end{priklad}
\res{$\frac{2}{3}$}
 
\item \bezlh
\begin{priklad}
\lim_{x \to 0} \frac{\sqrt{x^2+1} -1}{\sqrt{x^2+16} -4} 
\end{priklad}
\res{4}
 
\item \bezlh
\begin{priklad}
\lim_{x \to 1} \frac{x^2 - \sqrt{x}}{\sqrt{x} - 1} 
\end{priklad}
\res{3}
 
\item \bezlh
\begin{priklad}
\lim_{x \to 0} \frac{\sin{x} - \tg{x}}{\sin^3{x}}
\end{priklad}
\res{$-\frac{1}{2}$}
 
\item \bezlh
\begin{priklad}
\lim_{x \to -1} \frac{\sqrt[3]{1+2x} + 1}{ \sqrt[3]{2+x} + x}
\end{priklad}
\res{$\frac{1}{2}$}
 
\item \bezlh
\begin{priklad}
\lim_{x \to -8} \frac{\sqrt{1-x} - 3}{2+\sqrt[3]{x}} 
\end{priklad}
\res{-2}
 
\item \bezlh
\begin{priklad}
\lim_{x \to 0} \frac{\sqrt{1+x} - \sqrt{1-x}}{\sqrt[3]{1+x} - \sqrt[3]{1-x}} 
\end{priklad}
\res{$\frac{3}{2}$}
 
\item \bezlh
\begin{priklad}
\lim_{x \to -\infty} \frac{\sqrt[3]{x^3-2x^2}}{x+1} 
\end{priklad}
\res{1}
 
\item \bezlh
\begin{priklad}
\lim_{x \to 0} \frac{(1+x)(1+2x)(1+3x)-1}{x}
\end{priklad}
\res{$6$}
 
\item \bezlh
\begin{priklad}
\lim_{x \to 0} \frac{\sqrt{1-\cos(x^2)}}{1-\cos{x}}
\end{priklad}
\res{$\sqrt{2}$}
 
 
\odstavec{l'Hôpitalovo pravidlo povoleno}
 
\item 
\begin{priklad}
\lim_{x \to 0} \frac{\ln\cos x}{\ln\cos(\pi x)}
\end{priklad}
\res{$\frac{1}{\pi^2}$}
 
\item 
\begin{priklad}
\lim_{x \to 0} \frac{\cosh{x}-1}{\cos{x}-1}
\end{priklad}
\res{$-1$}
 
 
\item
\begin{priklad}
\lim_{x \to 0} \frac{\sinh(x)}{\sin{x}}
\end{priklad}
\res{$1$}
 
\item 
\begin{priklad}
\lim_{x \to -\infty} \frac{e^x - e^{-x}}{x+\frac{1}{x}}
\end{priklad}
\res{$+\infty$}
 
\item 
\begin{priklad}
\lim_{x \to 1} ~(1-x)\tg\left(x\frac{\pi}{2}\right)
\end{priklad}
\res{$\frac{2}{\pi}$}
 
\item 
\begin{priklad}
\lim_{x \to +\infty} \frac{\ln(4e^{-x})}{x}
\end{priklad}
\res{-1}
 
\item 
\begin{priklad}
\lim_{x \to +\infty} \sqrt{1+x+x^2} - \sqrt{1-x+x^2}
\end{priklad}
\res{1}
 
\item 
\begin{priklad}
\lim_{x \to -\infty} \sqrt{x^2+x+1} - \sqrt{x^2-x+1}
\end{priklad}
\res{-1}
 
\item 
\begin{priklad}
\lim_{x \to +\infty} \sqrt{x^2+x} - x 
\end{priklad}
\res{$\frac{1}{2}$}
 
\item 
\begin{priklad}
\lim_{x \to -\infty} \sqrt{x^2+x} - x 
\end{priklad}
\res{$+\infty$}
 
\item 
\begin{priklad}
\lim_{x \to 2} \big( \frac{1}{x-2} - \frac{1}{|x-2|} \big) 
\end{priklad}
\res{neex}
 
\item 
\begin{priklad}
\lim_{x \to 0_-} \frac{\cos{x}\sqrt{1-\cos{2x}}}{x} 
\end{priklad}
\res{$-\sqrt{2}$}
 
\item 
\begin{priklad}
\lim_{x \to 0} \frac{\sqrt{1-\cos{3x}}}{\sqrt{1-\cos^2{x}}}
\end{priklad}
\res{$\frac{3}{\sqrt{2}}$}
 
\item
\begin{priklad}
\lim_{x \to \frac{\pi}{4}} \tg{(2x)} \ln{(\tg{x})}
\end{priklad}
\res{$-1$}
 
\item 
\begin{priklad}
\lim_{x \to 0} \frac{\ln{\cos{x}}}{\ln{\cos{(2x)}}}
\end{priklad}
\res{$\frac{1}{4}$}
 
\item 
\begin{priklad}
\lim_{x \to 0} \frac{e^{x}-e^{2x}}{x}
\end{priklad}
\res{-1}
 
\item 
\begin{priklad}
\lim_{x \to +\infty} \sqrt{x^2+3x-1} - x
\end{priklad}
\res{$\frac{3}{2}$}
 
\item \bezlh
\begin{priklad}
\lim_{x \to +\infty} \sqrt{x+1} - \sqrt{x}
\end{priklad}
\res{0}
 
\item
\begin{priklad}
\lim_{x \to 1} \frac{\sqrt{4x^2-5x-x^3+2}}{x^2-1}
\end{priklad}
\res{neex}
 
\item 
\begin{priklad}
\lim_{x \to 1} \Big( \frac{1}{1-x} - \frac{3}{1-x^3}\Big)
\end{priklad}
\res{-1}
 
\item 
\begin{priklad}
\lim_{x \to +\infty} \frac{(1-x)(1-2x)(1-3x)(1-4x)(1-5x)}{(x-1)^5}
\end{priklad}
\res{$-5!$}
 
\item 
\begin{priklad}
\lim_{x \to +\infty} \left(\frac{2x^2-1}{2x^2+1} \right)^{\frac{x+2}{x+1}}
\end{priklad}
\res{$1$}
 
\item 
\begin{priklad}
\lim_{x \to +\infty} \left(\frac{3x^2-1}{2x^2+1} \right)^{\frac{1+2x}{x+1}}
\end{priklad}
\res{$\frac94$}
 
\item 
\begin{priklad}
\lim_{x \to +\infty} x \left(\ln(x+1)-\ln x \right)
\end{priklad}
\res{$1$}
 
\item
\begin{priklad}
\lim_{x \to +\infty} \arctg \frac{x}{\sqrt{1+x^2}}
\end{priklad}
\res{$\frac{\pi}{4}$}
 
 
\item
\begin{priklad}
\lim_{x \to -\infty} \arctg \frac{x}{\sqrt{1+x^2}}
\end{priklad}
\res{$-\frac{\pi}{4}$}
 
\item
\begin{priklad}
\lim_{x \to 1_-} \arctg \frac{1}{1-x}
\end{priklad}
\res{$\frac{\pi}{2}$}
 
 
\item 
\begin{priklad}
\lim_{x \to +\infty} \arcsin\frac{x}{x+1}
\end{priklad}
\res{$\frac{\pi}{2}$}
 
\odstavec{Spojitost}
 
\item Vyšetřete charakter bodů nespojitosti funkce
\begin{priklad}
f(x) = \arctg\frac{x^2-1}{x}
\end{priklad}
 
\res{v 0 skoková nespojitost}
 
 
\item Vyšetřete charakter bodů nespojitosti funkce
\begin{priklad}
f(x) = \sqrt{|x|}\arctg\frac{1}{x}
\end{priklad}
 
\res{v 0 odstranitelná nespojitost}
 
\item Vyšetřete charakter bodů nespojitosti funkce
\begin{priklad}
f(x) = x\sin\frac{1}{x}
\end{priklad}
 
\res{v 0 odstranitelná nespojitost}
 
 
\end{enumerate}