\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}
    Similarly, we find
    \begin{talign}
      |f(x) - 5| < \structure{0.1} \quad&\text{ whenever }\quad 0 < |x-3| < \alt<-2>{0.05}{\alert<-5>{\delta(0.1)}}\\
      |f(x) - 5| < \structure{0.01} \quad&\text{ whenever }\quad 0 < |x-3| < \alt<-3>{0.005}{\alert<-5>{\delta(0.01)}}\\
      \hphantom{1.1cm}|f(x) - 5| < \structure{0.001} \quad&\text{ whenever }\quad 0 < |x-3| < \alt<-4>{0.0005}{\alert<-5>{\delta(0.001)}}\hphantom{2cm}
    \end{talign}
    \pause
    The distances \structure{$0.1$},\; \structure{$0.01$}, \structure{\ldots} are called \emph{error tolerance}.
    \pause\medskip
    
    We have: \alert<-5>{$\delta(0.1) = 0.05$}\pause,\; \alert<-5>{$\delta(0.01) = 0.005$}\pause,\; \alert<-5>{$\delta(0.001) = 0.0005$}
    \pause\medskip
    
    Thus \alert{\emph{$\boldsymbol{\delta(\epsilon)}$ is a function of the error tolerance $\boldsymbol{\epsilon}$}}!
    \pause\bigskip
    
    We need to define $\delta(\epsilon)$ for arbitrary error tolerance $\epsilon > 0$:
    \begin{talign}
      |f(x) - 5| < \epsilon \quad&\text{ whenever }\quad 0 < |x-3| < \delta(\epsilon)
    \end{talign}\pause
    We want $|f(x) - 5| = 2|x-3| < \epsilon$. \pause We define \alert{$\delta(\epsilon) = \pause\epsilon/2$}.
  \end{exampleblock}
  \vspace{10cm}
\end{frame}