\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}