\begin{frame}
  \frametitle{Continuity: Function Composition}
  
  \begin{block}{}
    If $f$ is continuous at $b$ and $\lim_{x\to a} g(x) = b$, then
    \begin{talign}
      \lim_{x\to a} f(g(x)) = f(\lim_{x\to a}(g(x))
    \end{talign}
  \end{block}
  \pause\bigskip
  
  \begin{exampleblock}{}
    Evaluate $\lim_{x\to 4} \sin(\frac{\pi}{4+\sqrt{x}})$.
    \pause
    We have
    \begin{talign}
      \lim_{x\to 4} \sin(\frac{\pi}{4+\sqrt{x}}) 
      &\mpause[1]{= \sin(\lim_{x\to 4}  \frac{\pi}{4+\sqrt{x}})} &&\mpause[1]{\text{\textcolor{gray}{since $\sin$ is continuous}}}\\
      &\mpause[2]{= \sin(\frac{\pi}{4+\sqrt{4}})} &&\mpause[2]{\text{\textcolor{gray}{direct substitution}}}\\
      &\mpause[3]{= \sin(\frac{\pi}{6})} \mpause[4]{= \frac{1}{2}}
    \end{talign}
  \end{exampleblock}
\end{frame}