\begin{frame} \frametitle{Infinite One-Sided Limits} Like wise we define the one-sided infinite limits: \begin{itemize} \item[(a)] $\lim_{x \to a^-} f(x) = \infty$ \item[(b)] $\lim_{x \to a^-} f(x) = -\infty$ \item[(c)] $\lim_{x \to a^+} f(x) = \infty$ \item[(d)] $\lim_{x \to a^+} f(x) = -\infty$ \end{itemize} \begin{center} \scalebox{.45}{ (a) \begin{tikzpicture}[default,baseline=-1ex] \diagram{-1}{3}{-2}{2}{0} \diagramannotatez \draw[cblue,ultra thick] plot[smooth,domain=-.5:0.8,samples=20] function{(1/((x-1.3)**2)) -2} node [above] {$f(x)$}; \node [anchor=north,inner sep=1mm] at (1cm,-2cm) {$a$}; \draw [dashed,cred] (1cm,-2cm) -- (1cm,2cm); \end{tikzpicture} } \scalebox{.45}{ (b) \begin{tikzpicture}[default,baseline=-1ex] \diagram{-1}{3}{-2}{2}{0} \diagramannotatez \draw[cblue,ultra thick] plot[smooth,domain=-.5:0.8,samples=20] function{-(1/((x-1.3)**2)) +2} node [below left] {$f(x)$}; \node [anchor=north,inner sep=1mm] at (1cm,-2cm) {$a$}; \draw [dashed,cred] (1cm,-2cm) -- (1cm,2cm); \end{tikzpicture} } \scalebox{.45}{ (c) \begin{tikzpicture}[default,baseline=-1ex] \diagram{-1}{3}{-2}{2}{0} \diagramannotatez \draw[cblue,ultra thick] plot[smooth,domain=1.2:3,samples=20] function{(1/((x-0.7)**2)) -2} node [below] {$f(x)$}; \node [anchor=north,inner sep=1mm] at (1cm,-2cm) {$a$}; \draw [dashed,cred] (1cm,-2cm) -- (1cm,2cm); \end{tikzpicture} } \scalebox{.45}{ (d) \begin{tikzpicture}[default,baseline=-1ex] \diagram{-1}{3}{-2}{2}{0} \diagramannotatez \draw[cblue,ultra thick] plot[smooth,domain=1.2:3,samples=20] function{-(1/((x-0.7)**2)) +2} node [above] {$f(x)$}; \node [anchor=north,inner sep=1mm] at (1cm,-2cm) {$a$}; \draw [dashed,cred] (1cm,-2cm) -- (1cm,2cm); \end{tikzpicture} } \end{center} \pause \begin{alertblock}{} Note that $\infty$ and $-\infty$ are not considered numbers. \bigskip\pause If $\lim_{x \to a} f(x) = \infty$ then $\lim_{x \to a} f(x)$ does not exist.\\ It indicates a certain way in which the limit does not exist. \end{alertblock} % \begin{block}{Infinite Left-Limit} % \alert{$\lim_{x \to a^-} f(x) = \infty$} % if we can make the values of $f(x)$ arbitrarily large % by taking $x$ sufficiently close to $a$ with \alert{$x < a$}. % \end{block} % % \begin{block}{Infinite Right-Limit} % \alert{$\lim_{x \to a^+} f(x) = \infty$} % if we can make the values of $f(x)$ arbitrarily large % by taking $x$ sufficiently close to $a$ with \alert{$x > a$}. % \end{block} % % \begin{block}{Negative Infinite Left-Limit} % \alert{$\lim_{x \to a^-} f(x) = -\infty$} % if we can make the values of $f(x)$ arbitrarily large \alert{negative} % by taking $x$ sufficiently close to $a$ with \alert{$x < a$}. % \end{block} % % \begin{block}{Negative Infinite Right-Limit} % \alert{$\lim_{x \to a^+} f(x) = -\infty$} % if we can make the values of $f(x)$ arbitrarily large \alert{negative} % by taking $x$ sufficiently close to $a$ with \alert{$x > a$}. % \end{block} \end{frame}