\begin{frame} \frametitle{Infinite Limits} \begin{exampleblock}{} We consider the function $\frac{1}{x^2}$. What is $\lim_{x\to 0} \frac{1}{x^2}$ ? \end{exampleblock} \begin{center} \scalebox{.6}{ \begin{tikzpicture}[default] \diagram{-4}{4}{-.5}{4}{1} \diagramannotatez \diagramannotatex{-2,-1,1,2} \diagramannotatey{1} \draw[cred,ultra thick] plot[smooth,domain=-4:-0.5,samples=20] function{1/(x**2)}; \draw[cred,ultra thick] plot[smooth,domain=0.5:4,samples=20] function{1/(x**2)}; \end{tikzpicture} } \end{center} \pause As $x$ becomes close to $0$, $\frac{1}{x^2}$ becomes very large. \pause The values do not approach a number, so \alert{$\lim_{x\to 0} \frac{1}{x^2}$ does not exist!} \pause\bigskip Nevertheless, in this case, we write \begin{talign} \lim_{x\to 0} \frac{1}{x^2} = \infty \end{talign} to indicate that the values become larger and larger. \vspace{1cm} \end{frame}