\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 define \alert{$\delta(\epsilon) = \epsilon/2$}. Then the following holds \begin{talign} \alert{\text{if}\quad 0 < |x-3| < \delta(\epsilon) \quad\text{ then }\quad |f(x) - 5| < \epsilon} \end{talign} \pause In words this means: \begin{itemize} \item [] If $x$ is within a distance of $\epsilon/2$ from $3$ (and $x \ne 3$) \item [] then $f(x)$ is within a distance of $\epsilon$ from $5$. \end{itemize} \pause\medskip We can make $\epsilon$ arbitrarily small (but greater $0$),\\ and thereby make $f(x)$ arbitrarily close $5$. \end{exampleblock} \pause\bigskip This motivates the precise definition of limits\ldots \vspace{10cm} \end{frame}