\begin{frame} \frametitle{Limits at Infinity} \begin{exampleblock}{} Evaluate \begin{talign} \lim_{x\to \infty} (\sqrt{x^2 - 1} - x ) \end{talign} \pause We have \begin{talign} \lim_{x\to \infty} (\sqrt{x^2 - 1} - x ) &\mpause[1]{= \lim_{x\to \infty} \left( \frac{\sqrt{x^2 - 1} - x}{1} \cdot \frac{\sqrt{x^2 - 1} + x}{\sqrt{x^2 - 1} + x} \right) } \\ &\mpause[2]{= \lim_{x\to \infty} \frac{x^2 - 1 - x^2}{\sqrt{x^2 - 1} + x} } \\ &\mpause[3]{= \lim_{x\to \infty} -\frac{1}{\sqrt{x^2 - 1} + x} } \\ &\mpause[4]{= \lim_{x\to \infty} -\frac{1}{\sqrt{x^2 - 1} + x} \cdot \frac{\frac{1}{x}}{\frac{1}{x}} } \\ &\mpause[5]{= \lim_{x\to \infty} -\frac{\frac{1}{x}}{\sqrt{1 - \frac{1}{x^2}} + 1} } \mpause[6]{= \frac{0}{2} = 0} \end{talign} \end{exampleblock} \end{frame}