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\begin{document}
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\center{\Large 6. MATEMATIKA A1 FELADATSOR}

\begin{enumerate}
\item Határozza meg az alábbi sorozatok határértékeit:
\begin{enumerate}
\item $\displaystyle \lim_{n\to \infty }\left( 1+\frac{3}{2n+5}\right) ^{5n-7}$ \megold{$e^{15/2}$}
\item $\displaystyle \lim_{n\to \infty }\left( 1+\frac{2}{5n+5}\right) ^{3n+8}$ \megold{$e^{6/5}$}
\item $\displaystyle \lim_{n\to \infty }\left( \frac{n+3}{n+1}\right) ^{4n+9}$ \megold{$e^8$}
\item $\displaystyle \lim_{n\to \infty }\left( \frac{3n+3}{3n-1}\right) ^{\sqrt{n^2+4n+1}}$ \megold{$e^{4/3}$}
\item $\displaystyle \lim_{n\to \infty }\left( \frac{n^2+3n}{n^2+1}\right) ^{6n-9}$ \megold{$e^{18}$}
\item $\displaystyle \lim_{n\to \infty }\left( \frac{2^n+1}{2^n+3}\right) ^{2^{n+1}}$ \megold{$e^{-4}$}
\item $\displaystyle \lim_{n\to \infty }\left( \frac{4^n+1}{4^n+2^n}\right) ^{2^{n+1}}$ \megold{$e^{-2}$}
\item $\displaystyle \lim_{n\to \infty }\left( \frac{10^n+1}{10^n-2}\right) ^{10^{n-1}+n^2}$ \megold{$e^{3/10}$}
\item $\displaystyle \lim_{n\to \infty }\left( \frac{n^2}{n^2+2}\right) ^{3n^2-\lg n}$ \megold{$e^{-6}$}
\end{enumerate}
\item Deriválja az alábbi függvényeket:
\begin{enumerate}
\item $x^2+x^6-x^{11}$ \megold{$2x+6x^5-11x^{10}$}
\item $5x^2-4x+2$ \megold{$10x-8$}
\item $3x^2-4x^5+23x^7+11$ \megold{$6x-20x^4+161x^6$}
\item $78x^{42}-\frac{45}{x}$ \megold{$78 \cdot 42 x^41 + \frac{45}{x^2}$}
\item $(x^2+1)(x+1)$ \megold{$3x^2+2x+1$}
\item $\frac{3x^5-2x^3}{x^4}$ \megold{$3+\frac{2}{x^2}$}
\item $21\sqrt{x}-10\sqrt[3]{x}$ \megold{$21 \cdot \frac{1}{2\sqrt{x}} - 10 \cdot \frac{1}{3 \sqrt[3]{x^2}}$}
\item $\frac{2x^6-1}{\sqrt{x}}$ \megold{$11 x^{9/2} + \frac{1}{2} x^{-3/2}$}
\end{enumerate}
\item Határozza meg az $f(x)$ függvény $x_0$-ban vett érintőjét
\begin{enumerate}
\item $x^2$, $x_0=3$ \megold{$y=6x-9$}
\item $\sqrt{x}$, $x_0=9$ \megold{$y = \frac{1}{6}x + \frac{3}{2}$}
\item $x^2-x$, $x_0=2$ \megold{$y=3x-4$}
\item $\frac{1}{x}$, $x_0=5$ \megold{$y = - \frac{1}{25} x + \frac{2}{5}$}
\item $x^3$, $x_0=1$ \megold{$y=3x -2$}
\end{enumerate}

\end{enumerate}


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