\begin{frame}
  \frametitle{One-Sided Limits (From the Left)}

  \begin{minipage}{.5\textwidth}
  \begin{exampleblock}{}
    The 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}
  \end{exampleblock}\vspace{.5ex}
  \end{minipage}~\quad~
  \begin{minipage}{.49\textwidth}
    {\def\diax{t}\def\diay{H(t)}
    \scalebox{.6}{
    \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{minipage}
  \medskip\pause
  
  $H(t)$ approaches $0$ as $t$ approaches $0$ from the \alert{left}.
  We write:
  \begin{align*}
    \lim_{t\to0^{\alert{\boldsymbol{-}}}} H(t) = 0
  \end{align*}
  The symbol $t\to0^{-}$ indicates that we consider only values $t < 0$.\hspace*{-2ex}
  \bigskip\pause
  
  \begin{block}{}
    We write \hspace{2cm} $\lim_{x\to a^{\alert{-}}} f(x) = L$ \hspace{2cm} and say
    \begin{talign}
      &\text{``the \alert{left-hand} limit of $f(x)$, as $x$ approaches $a$, is $L$''}, or \\
      &\text{``the limit of $f(x)$, as $x$ approaches $a$ \alert{from the left}, is $L$''}
    \end{talign}
    if we can make the values $f(x)$ arbitrarily close to $L$ by taking $x$ sufficiently close to $a$ and $\alert{x < a}$.
  \end{block}
\end{frame}