\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}