\begin{frame} \frametitle{Infinite Limits: Examples} \begin{exampleblock}{} Find \begin{talign} \lim_{x \to 3^-} \frac{2x}{x-3} &&\text{and}&& \lim_{x \to 3^+} \frac{2x}{x-3} \end{talign} \end{exampleblock} \pause\smallskip \begin{minipage}{1\textwidth} \begin{itemize} \item [] $\lim_{x \to 3^-} \frac{2x}{x-3} = \alt<-2>{?}{-\infty}$ \onslide<-2>{\choice{a} $0$ \choice{b} $1$ \choice{c} $\infty$ \choice{d} $-\infty$} \pause\pause \item [] $\lim_{x \to 3^+} \frac{2x}{x-3} = \alt<-4>{?}{\infty}$ \onslide<-4>{\choice{a} $0$ \choice{b} $1$ \choice{c} $\infty$ \choice{d} $-\infty$} \end{itemize} \end{minipage} % \hspace*{-.5\textwidth}\begin{minipage}{.49\textwidth} \onslide<12->{\scalebox{.6}{ \begin{tikzpicture}[default,baseline=-1ex] \diagram{-1}{5}{-2}{2}{0} \diagramannotatez \diagramannotatex{1,2,3,4} \node at (0,1cm) [anchor=east,inner sep=1mm] {5}; \draw[cblue,ultra thick] plot[smooth,domain=-1:2.5,samples=20] function{(2*x)/(x-3)/5}; \draw[cblue,ultra thick] plot[smooth,domain=3.6:5,samples=20] function{(2*x)/(x-3)/5} node [below] {$f(x)$}; \draw [dashed,cred] (3cm,-2cm) -- (3cm,2cm); \end{tikzpicture} }} \end{minipage} \pause\pause\smallskip If $x$ is close to $3$ and $x < 3$ (approaching from the left), then: \begin{itemize} \pause \item $2x$ is close to $6$, \pause \item $x-3$ is a small negative number, \pause \item and thus $2x/(x-3)$ is a large negative number. \end{itemize} \pause Hence $\lim_{x \to 3^-} \frac{2x}{x-3} = -\infty$. \pause\smallskip Similarly for $x$ close to $3$ and $x > 3$, but now $x-3$ is positive. % But now $x-3$ is a small positive number, % and hence $\lim_{x \to 3^+} \frac{2x}{x-3} = \infty$. \end{frame}