\begin{frame} \frametitle{Limits at Infinity: Laws} \begin{block}{} All \emph{limits laws} for $\lim_{x\to a}$ work also for $\lim_{x\to \pm\infty}$, \alert{except for}: \begin{talign} \lim_{x\to a} x^n = a^n && \lim_{x\to a} \sqrt[n]{x} = \sqrt[n]{a} \end{talign} \end{block} \pause\bigskip For example, we can derive the following important theorem: \begin{block}{} For $r > 0$ we have \begin{talign} \lim_{x\to \infty} \frac{1}{x^r} = 0 \end{talign} \pause and if $x^r$ is defined for all $x$, then also \begin{talign} \lim_{x\to -\infty} \frac{1}{x^r} = 0 \end{talign} \end{block} \pause \begin{exampleblock}{Proof} \vspace{-1ex} \begin{talign} \lim_{x\to \infty} \frac{1}{x^r} \mpause[1]{= \lim_{x\to \infty} (\frac{1}{x})^r } \mpause[2]{= (\lim_{x\to \infty} \frac{1}{x})^r } \mpause[3]{= 0^r} \mpause[4]{= 0} \end{talign} \end{exampleblock} \end{frame}