\begin{frame}
  \frametitle{Infinite Limits at Infinity: Heuristics}
  
  \alert{All on this slide is heuristics, not laws!}
  \pause\bigskip
  
  On the last slide we could have reasoned as follows:
  \begin{talign}
    \lim_{x\to\infty} (x^2 - x) 
    \mpause[1]{= \lim_{x\to\infty} x \cdot \lim_{x\to\infty} (x - 1) }
    \mpause[2]{= \infty \cdot \infty}
    \mpause[3]{= \infty}
  \end{talign}
  \pause\pause\pause\pause\bigskip

  Valid calculations with $\infty$ and $x$ a real number:
  \begin{talign}
    \infty + \infty = \infty &&
    \infty + x = \infty &&
    \frac{x}{\infty} = 0 
  \end{talign}
  \begin{talign}
    \frac{\infty}{x} = \infty \text{\;\;if $x > 0$} &&
    \frac{\infty}{x} = -\infty \text{\;\;if $x < 0$}
  \end{talign}
  \pause\smallskip
  
  \alert{Invalid, undefined expressions}:
  \begin{talign}
    \infty - \infty &&
    \infty + (-\infty) &&
    \frac{\infty}{\infty} &&
    0 \cdot \infty
  \end{talign}
\end{frame}