\begin{frame} \frametitle{Calculating Limits using Limit Laws} \begin{block}{} \begin{enumerate} \item $\lim_{x\to a} \;[f(x) + g(x)] = \lim_{x\to a} f(x) + \lim_{x\to a} g(x)$ \item $\lim_{x\to a} \;[f(x) - g(x)] = \lim_{x\to a} f(x) - \lim_{x\to a} g(x)$ \item $\lim_{x\to a} \;[c \cdot f(x)] = c \cdot \lim_{x\to a} f(x)$ \item $\lim_{x\to a} \;[f(x) \cdot g(x)] = \lim_{x\to a} f(x) \cdot \lim_{x\to a} g(x)$ \item $\lim_{x\to a} \frac{f(x)}{g(x)} = \frac{\lim_{x\to a} f(x)}{\lim_{x\to a} g(x)}$ if $\lim_{x\to a} g(x) \ne 0$ \end{enumerate} \end{block} \bigskip \begin{indentation}{-.2cm}{-1cm} \begin{minipage}{.44\textwidth} \scalebox{.6}{ \begin{tikzpicture}[default] \diagram{-3}{3}{-2}{3}{1} \diagramannotatez \diagramannotatex{-3,-2,-1,1,2} \diagramannotatey{-1,1,2} \draw[cred,ultra thick] plot[smooth,domain=-3:-2,samples=20] function{2-(x+2)**2}; \draw[cred,ultra thick] (-2,2) to[out=-45,in=180,looseness=2] (1,2) to[out=0,in=135,looseness=1] node [at end,below] {$f(x)$} (3,.5); \draw[cblue,ultra thick] (-3,-2/3) to (1,-2); \draw[cblue,ultra thick] (1,-1) to node[at end,above] {$g(x)$} (3,1); \node[exclude={cred}] at (-2,2) {}; \node[include={cred}] at (-2,3) {}; \node[exclude={cblue}] at (1,-1) {}; \node[include={cblue}] at (1,-2) {}; \end{tikzpicture} } \end{minipage} \begin{minipage}{.65\textwidth} \pause Use these graphs to estimate:\\[-1ex] \pause \begin{overlayarea}{\textwidth}{3.5cm}% \only<-7>{% \begin{itemize} \item[1.] $\lim_{x\to -2}\; [f(x) + 5g(x)]$\\ \pause $= \lim_{x\to -2} f(x) + \lim_{x\to -2} \;[5g(x)]$\\ \pause $= \lim_{x\to -2} f(x) + 5\, \lim_{x\to -2} g(x)$\\ \pause $= 2 + 5(-1)$\\ \pause $= -3$ \end{itemize} } \pause[8] \only<8-11>{% \begin{itemize} \item[2.] $\lim_{x\to 1}\; [f(x)g(x)]$\\\pause $= \lim_{x\to 1} f(x) \cdot \lim_{x\to 1} g(x)$\\\pause\smallskip \flash $\lim_{x\to 1} g(x)$ does not exist\\[1ex]\pause (we cannot use the limit laws) \end{itemize} } \pause[12] \only<12-20>{% \begin{itemize} \item[2a.] $\lim_{x\to 1^-}\; [f(x)g(x)]$\\\pause $= \lim_{x\to 1^-} f(x) \cdot \lim_{x\to 1^-} g(x)$\\\pause $= 2 \cdot -2 $\pause $= -4$\pause \item[2b.] $\lim_{x\to 1^+}\; [f(x)g(x)]$\\\pause $= \lim_{x\to 1^+} f(x) \cdot \lim_{x\to 1^+} g(x)$\\\pause $= 2 \cdot -1 $\pause $= -2$\pause \item[$\implies$] $\lim_{x\to 1}\; [f(x)g(x)]$ does not exist \end{itemize} } \pause[21] \only<21-24>{% \begin{itemize} \item[3.] $\lim_{x\to 2}\; \frac{f(x)}{g(x)}$\pause \ $ = \frac{\lim_{x\to 2} f(x)}{\lim_{x\to 2} g(x)}$\\\pause \flash $\lim_{x\to 2} g(x) = 0$\\[1ex]\pause (we cannot use the limit laws) \end{itemize} } \pause[25] \only<25-30>{% \noindent Lets try without limit laws: \begin{itemize} \item[3a.] $\lim_{x\to 2^-}\; \frac{f(x)}{g(x)}$\pause \ $ = -\infty$\\\pause since $\lim_{x\to 2^-} f(x) \approx 1.6$, and\\ $g(x)$ approaches $0$, $g(x) < 0$\pause \item[3b.] $\lim_{x\to 2^+}\; \frac{f(x)}{g(x)}$\pause \ $ = \infty$\\\pause since $\lim_{x\to 2^+} f(x) \approx 1.6$, and\\ $g(x)$ approaches $0$, $g(x) > 0$ \end{itemize} } \end{overlayarea} \end{minipage} \end{indentation} \end{frame}