\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} }\quad\quad \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}