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\begin{frame}
\frametitle{Precise Definition of Limits: Example}

\begin{exampleblock}{}
\vspace{-1ex}
\begin{talign}
f(x) = \begin{cases}
2x - 1 &\text{for $x \ne 3$}\\
6 &\text{for $x = 3$}
\end{cases}
\end{talign}
We have derived
\begin{talign}
|f(x) - 5| < 0.1 \quad\text{ whenever }\quad 0 < |x-3| < 0.05
\end{talign}
\pause
In words this means:
\begin{itemize}
\item [] If $x$ is within a distance of $0.05$ from $3$ (and $x \ne 3$)
\item [] then $f(x)$ is within a distance of $0.1$ from $5$.
\end{itemize}
\end{exampleblock}
\vspace{10cm}
\end{frame}