\begin{frame}
  \frametitle{L'Hospital's Rule}

  \begin{block}{}
    A limit of the form
    \begin{talign}
      \lim_{x \to a} [f(x)]^{g(x)} 
    \end{talign}
    is an indeterminate form 
    \begin{itemize}
      \item \emph{of type $0^0$} \quad if \quad $\lim_{x \to a} f(x) = 0$ and $\lim_{x \to a} g(x) = 0$
      \item \emph{of type $\infty^0$} \quad if \quad $\lim_{x \to a} f(x) = \infty$ and $\lim_{x \to a} g(x) = 0$
      \item \emph{of type $1^\infty$} \quad if \quad $\lim_{x \to a} f(x) = 1$ and $\lim_{x \to a} g(x) = \infty$
    \end{itemize}
  \end{block}
  \pause\medskip

  Each of these cases can be treated by writing the limit as:
  \begin{talign}
    \lim_{x \to a} [f(x)]^{g(x)} 
    &= \lim_{x \to a} e^{\ln \left( [f(x)]^{g(x)}  \right)} \\
    &\mpause[1]{= \lim_{x \to a} e^{g(x) \ln f(x)} }
    \mpause[2]{= e^{\lim_{x \to a} \left( g(x) \ln f(x) \right)}}
  \end{talign}
  \pause\pause\pause
  
  \begin{alertblock}{}
    Other types are \emph{not} indeterminate forms: $0^\infty$, $1^0$ and $\infty^1$.
  \end{alertblock}
\end{frame}