\begin{frame} \frametitle{Limit: Examples} \begin{exampleblock}{} Guess the value of \begin{talign} \lim_{x\to 0} \sin \frac{\pi}{x} \end{talign} \end{exampleblock} \pause \smallskip \begin{center} \begin{minipage}{.29\textwidth} \scalebox{.9}{\small \begin{tabular}{|l|l|} \hline $x$ & $f(x)$ \\ \hline $\pm 1$ & $0$ \\ \hline $\pm 0.1$ & $0$ \\ \hline $\pm 0.01$ & $0$ \\ \hline $\pm 0.001$ & $0$ \\ \hline \end{tabular} } \end{minipage}~~~ \begin{minipage}{.59\textwidth} \pause This suggest that the limit is $0$. \bigskip \pause However, this is \alert{wrong}: \end{minipage} \end{center} \begin{center} \scalebox{.6}{ \begin{tikzpicture}[default] \diagram{-6}{6}{-1}{1.5}{1} \diagramannotatez \diagramannotatex{-1,1} \diagramannotatey{1} \draw[cblue] plot[smooth,domain=-6:6,samples=2000] function{sin(pi/ x)}; \def\x{0} \def\y{{1}} \node[exclude={cblue}] at (\x,\y) {}; \end{tikzpicture} } \end{center} \vspace{-1ex} \pause \begin{exampleblock}{} $\sin(\frac{\pi}{x}) = 0$ for arbitrarily small $x$, but also\\ \pause $\sin(\frac{\pi}{x}) = 1$ for arbitrarily small $x$; \pause e.g. $x = \frac{1}{2.5}$, \pause $\frac{1}{4.5}$, \pause $\frac{1}{6.5}$,\ldots\\[.5ex] \pause Hence: \alert{The limit $\lim_{x\to 0} \sin \frac{\pi}{x}$ does not exist.} \end{exampleblock} \vspace{10cm} \end{frame}