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\begin{frame}
\frametitle{Limit: Caution with Calculators}

\begin{exampleblock}{}
Guess the value of
\begin{talign}
\lim_{x\to 0} \frac{\sqrt{x^2 + 9} -3}{x^2}
\end{talign}
\end{exampleblock}

\onslide<6->{
\begin{center}
\scalebox{.7}{
\begin{tikzpicture}[default,yscale=2,xscale=.5]
{\def\diaborderx{1cm}
\def\diabordery{.3cm}
\diagram[1]{-5}{5}{0}{1.2}{1}}
\diagramannotatex{0,1,2,3}
\diagramannotatey{0.5,1}
\draw[cgreen,ultra thick] plot[smooth,domain=-5:5,samples=20] function{(sqrt(x**2 + 9) -3) / (x**2)};

\def\x{0}
\def\y{{.166666666}}
\node[exclude={cgreen}] at (\x,\y) {};
\end{tikzpicture}
\scalebox{.7}{
\begin{tikzpicture}[default,scale=5]
{\def\diaborderx{.1cm}
\def\diabordery{.1cm}
\diagram[.1]{-.5}{.5}{0}{.5}{1}}
\diagramannotatex{0,0.2,0.4}
\diagramannotatey{0.1,0.3}
\draw[cgreen,ultra thick] plot[smooth,domain=-.5:.5,samples=20] function{(sqrt(x**2 + 9) -3) / (x**2)};

\def\x{0}
\def\y{{.166666666}}
\node[exclude={cgreen}] at (\x,\y) {};
\end{tikzpicture}
}
\end{center}
}
\pause

\begin{minipage}{.37\textwidth}
{\small
\begin{tabular}{|l|l|}
\hline
$x$ & $f(x)$ \\
\hline
$\pm 1.0$ & $0.16228$ \\
\hline
$\pm 0.5$ & $0.16553$ \\
\hline
$\pm 0.1$ & $0.16662$ \\
\hline
$\pm 0.01$ & $0.16667$ \\
\hline
$\pm 0.0001$ & $0.20000$ \\
\hline
$\pm 0.00001$ & $0.00000$ \\
\hline
$\pm 0.000001$ & $0.00000$ \\
\hline
\end{tabular}
}
\end{minipage}~~~
\begin{minipage}{.62\textwidth}
\pause
Is the limit $0$? \pause \alert{NO}
\medskip\pause

Problem: calculator gives wrong values!
%     The calculator has only a fixed number of digits, and needs to round.
For small $x$ it rounds $\sqrt{x^2+9} - 3$ to $0$.
\pause\medskip

\begin{exampleblock}{}
The correct limit is $\frac{1}{6} = 0.166666 \ldots$
\end{exampleblock}
\end{minipage}

\vspace{10cm}
\end{frame}