\begin{frame} \frametitle{L'Hospital's Rule} \begin{block}{} A limit of the form \begin{talign} \lim_{x \to a} \frac{f(x)}{g(x)} \end{talign} where both \begin{talign} \lim_{x \to a} f(x) = 0 &&\text{and}&& \lim_{x \to a} g(x) = 0 \end{talign} is called \emph{indeterminate form of type $\frac{0}{0}$}. \end{block} \pause Often cancellation of common factors helps: \begin{talign} \lim_{x\to 1} \frac{x^2 - x}{x^2 - 1} \mpause[1]{= \lim_{x\to 1} \frac{(x-1)x}{(x-1)(x+1)}} \mpause[2]{= \lim_{x\to 1} \frac{x}{x+1}} \mpause[3]{= \frac{1}{2}} \end{talign} \pause\pause\pause But not for examples like: \begin{exampleblock}{} \begin{talign} \lim_{x\to 0} \frac{\sin x}{x} &&\text{and}&& \lim_{x\to 1} \frac{\ln x}{x-1} \end{talign} \end{exampleblock} \end{frame}