\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}