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\usepackage{tikz}

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\begin{document}

Consider the following piecewise function: 

\[f(x) = \begin{cases}
 2+x & x \leq -3 \\
 -\frac{1}{4}(x+1)^2  & -3 < x < -1 \\
 \frac{2}{1-x} - 1 & -1 < x < 1 \\
  2-x & 1 \leq x < 4 \\
  -2 & x=4 \\
  3 & x > 4 \\	
\end{cases}
\]

whose graph is shown here: 

\begin{center}
     \begin{tikzpicture}[scale=0.7]
    \draw[black, thick, <->] (-5.5, 0) -- (5.5, 0);
    \foreach \y in {-5, ..., 5}
    \draw[gray] (-5, \y) -- (5,\y);
    \draw[black, thick, <->] (0, -5.5) -- (0, 5.5);
    \foreach \x in {-5, ..., 5}
    \draw[gray] (\x, -5) -- (\x,5);
    
    \draw[black, very thick, domain=-5:-3] plot(\x, 2+\x);
    \draw[fill, black] (-3,-1) circle[radius=0.1];
    \draw[black, very thick, domain=-3:-1] plot(\x, {-1*(\x+1)*(\x+1)/4});
    \draw[black] (-1,0) circle[radius=0.1];
    \draw[black] (-1,0) circle[radius=0.1];
    % \draw[fill, black] (-2, -4) circle[radius=0.1];
    \draw[black, very thick, domain=-1:0.68] plot(\x, {2/(1- \x) - 1});
    \draw[fill, black] (1,1) circle[radius=0.1];
    \draw[black, very thick, domain=1:4] plot (\x, 2-\x);
    \draw[fill, black] (4,-2) circle[radius=0.1];
    \draw[black] (4,3) circle[radius=0.1];
    \draw[black, very thick, domain=4:5] plot(\x, 3);
    % \draw[black] (3,2) circle[radius=0.1];
    \end{tikzpicture}
    \end{center}

\begin{enumerate}
\item There is a table on the last page of the workshop. Fill out all of the information requested in it. You will be asked about one-sided limits, full limits, and continuity of this function $f$. Do whatever scratch work you need, but you only need to write the answers in the table. 
\item Come up with a new function $g(x)$, defined piecewise, with the following properties:
\begin{itemize}
\item $g$ has a jump discontinuity at $x=-2$,
\item $g$ has a removable discontinuity at $x=1$,
\item $g$ has an infinite discontinuity at $x=3$, and
\item $g$ is continuous everywhere else.
\end{itemize}
\end{enumerate}


\newpage

\renewcommand{\arraystretch}{3}

\begin{tabular}{|c|c|c|c|c|}\hline
    & {\Large $\ \ \ c = -3\ \ \ $} & {\Large $\ \ \ c = -1\ \ \ $} & {\Large $\ \ \ c=1\ \ \ $} & {\Large $\ \ \ c=4\ \ \ $} \\\hline
     \Large{$\displaystyle \lim_{x\rightarrow c^+} f(x)$} & & & &  \\[15pt]\hline
     \Large{$\displaystyle \lim_{x\rightarrow c^-} f(x)$} & & & &  \\[15pt]\hline
     \Large{$\displaystyle \lim_{x\rightarrow c} f(x)$} & & & &  \\[15pt]\hline
     \Large{$f(c)$} & & & &  \\[15pt]\hline
     \Large{Is $f$ continuous at c?} & & & &  \\[15pt]\hline
\end{tabular}


\end{document}

