\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}