44/154
\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

\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}