\begin{frame} \frametitle{Limits at Infinity: Horizontal Asymptotes} \begin{block}{} The line $y = L$ is called \emph{horizontal asymptote} of a function $f$ if \begin{talign} \lim_{x\to\infty} f(x) = L &&\text{or} &&\lim_{x\to-\infty} f(x) = L \end{talign} \end{block} \pause\bigskip \only<2>{ \begin{exampleblock}{} The function $f(x) = \frac{x^2 - 1}{x^2 + 1}$ has a horizontal asymptote at $y = 1$. \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} \end{exampleblock} } \only<3>{ \begin{exampleblock}{} The inverse tangent $\tan^{-1}$ has horizontal asymptotes \begin{talign} y = -\frac{\pi}{2} && \text{and} && y = \frac{\pi}{2} \end{talign} \begin{center} \scalebox{.7}{ \begin{tikzpicture}[default,yscale=.8] \diagram{-7}{7}{-2}{2}{1} \diagramannotatez \diagramannotatexx{-pi/$-\pi$,pi/$\pi$} \diagramannotateyy{{-0.5*pi}/$-\frac{\pi}{2}$,{0.5*pi}/$\frac{\pi}{2}$} \draw[cgreen,ultra thick] plot[smooth,domain=-7:7,samples=50] function{atan(x)}; \onslide<2->{ \draw[cred,dashed] (-7,-0.5*pi) -- (7,-0.5*pi); \draw[cred,dashed] (-7,0.5*pi) -- (7,0.5*pi); } \end{tikzpicture} } \end{center}\vspace{-1ex} \pause \begin{talign} \lim_{x\to -\infty} \tan^{-1}{x} = -\frac{\pi}{2} && \lim_{x\to \infty} \tan^{-1}{x} = \frac{\pi}{2} \end{talign} \end{exampleblock} } \vspace{10cm} \end{frame}