\begin{frame} \frametitle{1st Midterm Exam - Review} \begin{exampleblock}{} Prove or disprove that the following limit exists \begin{talign} \lim_{x\to 5} \frac{x-5}{|x-5|} \end{talign} \pause For $x < 5$ we have $\frac{x-5}{|x-5|} = -1$. \pause Thus \begin{talign} \lim_{x\to 5^-} \frac{x-5}{|x-5|} &\mpause[1]{= \lim_{x\to 5^-} -1} \mpause[2]{= -1} \end{talign} \pause\pause\pause For $x > 5$ we have $\frac{x-5}{|x-5|} = 1$. \pause Thus \begin{talign} \lim_{x\to 5^+} \frac{x-5}{|x-5|} &\mpause[1]{= \lim_{x\to 5^+} 1} \mpause[2]{= 1} \end{talign} \pause\pause\pause\medskip The limit $\lim_{x\to 5} \frac{x-5}{|x-5|}$ does not exist since the left- and the right-limit are different. \end{exampleblock} \end{frame}