\begin{frame} \frametitle{Limit: Examples} \begin{exampleblock}{} The Heaviside function $H$ is defined by \begin{talign} H(t) = \begin{cases} 0 & \text{if $t < 0$}\\ 1 & \text{if $t \ge 0$}\\ \end{cases} \end{talign} What is $\lim_{t\to 0} H(t)$? \end{exampleblock} \smallskip \begin{center} {\def\diax{t}\def\diay{H(t)} \scalebox{.7}{ \begin{tikzpicture}[default] \diagram{-3}{3}{-1}{1.5}{1} \diagramannotatez \diagramannotatex{-1,1} \diagramannotatey{1} \draw[cred,ultra thick] plot[smooth,domain=-3:0,samples=20] function{0}; \draw[cred,ultra thick] plot[smooth,domain=0:3,samples=20] function{1}; \def\x{0} \def\y{{1}} \node[include={cred}] at (\x,\y) {}; \node[exclude={cred}] at (0,0) {}; \end{tikzpicture} }} \end{center}\vspace{-1ex} \begin{itemize} \pause \item As $t$ approaches $0$ from the left, $H(t)$ approaches $0$. \pause \item As $t$ approaches $0$ from the right, $H(t)$ approaches $1$. \end{itemize} \medskip\pause Thus there is not single number that $H(t)$ approaches.\vspace{-.5ex} \begin{exampleblock}{} The limit $\lim_{t\to 0} H(t)$ does not exist. \end{exampleblock} \vspace{10cm} \end{frame}