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\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}