\documentclass [12pt] {article}

\usepackage{graphicx, xypic, amsmath}

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\textheight 8.9in
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\textwidth 6.5in

\begin{document}
\begin{center}

\Huge{Continuity}\\
\end{center}

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


\begin{enumerate}
\item Consider the piecewise-defined function 

\hspace{2in} $f(x)=\begin{cases}
x^2-3 & \text{ if } -1\leq x\\
\displaystyle{\frac{2x^2+2x}{x+1}} & \text{ if } x< -1\\
\end{cases}$

\begin{enumerate}
\item Draw the graph of $f$ on the axis below.
\end{enumerate}

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

\item Find $\displaystyle{\lim_{x\rightarrow -1} f(x)}$ either numerically or graphically.

\vskip 1in

\item Find $f(-1)$
\vskip .5in
\item Is $f$ continuous at $x=-1$?



\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 Is $g$ continuous at:

$x=0$ \hspace{1.5in} $x=2$ \hspace{1.5in} $x=3$?
\vskip .4in
\item Find all the points that $g$ is discontinuous.
\vskip 1in

\end{enumerate}

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

$\alpha(-1)=-3,$ \hspace{.01in} $\displaystyle{\lim_{x\rightarrow -1}\alpha(x)=2}$, \hspace{.01in} 
$\alpha$ is continuous on $(0,3)$, \hspace{.01in}  and not continuous at $x=4$.


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





\end{enumerate}
\end{document}
