\begin{frame} \frametitle{Limits at Infinity} Lets investigate the behavior of the function \begin{talign} f(x) = \frac{x^2 - 1}{x^2 + 1} \end{talign} when $x$ becomes large: \begin{center} \scalebox{.7}{ \begin{tikzpicture}[default] \diagram{-7}{7}{-1.2}{1.3}{1} \diagramannotatez \diagramannotatex{-1,1} \diagramannotatey{1} \draw[cgreen,ultra thick] plot[smooth,domain=-7:7,samples=20] function{(x**2 - 1)/(x**2 + 1)}; \onslide<2->{ \draw[cred,dashed] (-7,1) -- (7,1); } \end{tikzpicture} } \end{center} \pause As $x$ grows larger, the values of $f(x)$ get closer and closer to $1$.\\ \pause\medskip This is expressed by \begin{talign} \lim_{x\to\infty} \frac{x^2 - 1}{x^2 + 1} = 1 \end{talign} \end{frame}