\begin{frame} \frametitle{One-Sided Limits (From the Right)} \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 $1$ as $t$ approaches $0$ from the \alert{right}. We write: \begin{align*} \lim_{t\to0^{\alert{\boldsymbol{+}}}} H(t) = 1 \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{right-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 right}, 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}