\documentclass [12pt] {article}

\usepackage{graphicx, xypic, amsmath}

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\textheight 8.9in
\oddsidemargin 0pt
\evensidemargin \oddsidemargin
\marginparwidth 0.5in
\textwidth 6.5in

\begin{document}
\begin{center}

\Huge{Limits}\\
\end{center}

\normalsize
Note: a personal copy of this worksheet may be used during the quiz on Tuesday.


\begin{enumerate}
\item Consider $\displaystyle{f(x)=\frac{2x^2+4x}{x+2}}$.
\begin{enumerate}
\item Fill in the following table:

\vskip .2in
\begin{tabular*}{6in}{c|@{\extracolsep{\fill}}c@{\extracolsep{\fill}}c@{\extracolsep{\fill}}c@{\extracolsep{\fill}}c@{\extracolsep{\fill}}c@{\extracolsep{\fill}}c@{\extracolsep{\fill}}c}
x &-1 &-1.9 &-1.99 &-2 &-2.01 &-2.1 &-3\\
\hline
f(x) &\\ 
\end{tabular*}

\vskip .2in

\item Use the data above to find $\displaystyle{\lim_{x\rightarrow -2} f(x)}$.
\vskip .5in
\item Draw the graph of $f$ on the axis below and verify your answer.
\end{enumerate}

\begin{center}
\includegraphics[height=3.5in]{blank_5}	
\end{center}


\newpage 
\item Let $g$ be the piece-wise defined function below.  This means the graph of $g$ is the \emph{entire} dotted graph shown below. 

\begin{center}
\includegraphics[height=2in]{Piece4Limits}	
\end{center}

\begin{enumerate}
\item Find the following \emph{if} it exits:

$\displaystyle{\lim_{x\rightarrow 1} g(x)}$ 
\hspace{1.5in} $\displaystyle{\lim_{x\rightarrow 0} g(x)}$
\hspace{1.5in} $\displaystyle{\lim_{x\rightarrow 3} g(x)}$
\vfill

$\displaystyle{\lim_{x\rightarrow 2} g(x)}$
\hspace{1.5in}$\displaystyle{\lim_{x\rightarrow -2} g(x)}$
\hspace{1.5in}$\displaystyle{\lim_{x\rightarrow 4} g(x)}$
\vfill
\end{enumerate}

\item Sketch a graph of a function $\alpha$ that satisfies \emph{all} of the following:

$\alpha(-1)=2$. \hspace{.1in} $\displaystyle{\lim_{x\rightarrow -1}\alpha(x)=-3}$, \hspace{.1in} 
$\displaystyle{\lim_{x\rightarrow 2}\alpha(x)=\infty}$, \hspace{.1in}  and the $\displaystyle{\lim_{x\rightarrow 4}\alpha(x)}$ does not exist.


\begin{center}
\includegraphics[height=2.5in]{blank_5}	
\end{center}





\end{enumerate}
\end{document}
