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\title{Graphs and limits}
\date{September 1, 2022}

\begin{document}
\ \vspace{-.3in}

\maketitle

\vspace{-.2in}

\prob{
Sketch a graph of the function, then use the graph to help find the limits.

\begin{enumerate}[a)]

\item $\displaystyle f(x) = \begin{cases} -x &\text{ if } x < 0 \\  x &\text{ if }  x > 0 \end{cases}$

\noindent \begin{tikzpicture}
\draw[step=0.6cm,color=lightgray,very thin] (-4,-4) grid (4,4);
\end{tikzpicture}

\item $\displaystyle \lim_{x \to 0} f(x) = $
\item Is $f(0)$ defined?
\end{enumerate} 
}


\prob{
Sketch a graph of the function, then use the graph to answer questions about the limits of the function.

\begin{enumerate}[a)]

\item $\displaystyle f(x) = \begin{cases} (x+1)^2 &\text{ if } x \leq 0 \\  x^2 + 1 &\text{ if } x > 0 \end{cases}$

\noindent \begin{tikzpicture}
\draw[step=0.6cm,color=lightgray,very thin] (-4,-4) grid (4,4);
\end{tikzpicture}

\item $\displaystyle \lim_{x \to -1} f(x) = $

\item $\displaystyle \lim_{x \to 0^-} f(x) = $

\item $\displaystyle \lim_{x \to 0^+} f(x) = $

\end{enumerate} 
}


\newpage

\prob{
Sketch a graph of the function, then use the graph to answer questions about the limits of the function.

\begin{enumerate}[a)]

\item $\displaystyle f(x) = \begin{cases} 1 - x &\text{ if } x< 1 \\  (x-2)^2 &\text{ if }  x \geq 1 \end{cases}$

\noindent \begin{tikzpicture}
\draw[step=0.6cm,color=lightgray,very thin] (-4,-4) grid (4,4);
\end{tikzpicture}

\item $\displaystyle \lim_{x \to 0} f(x) = $
\item $\displaystyle \lim_{x \to 1^-} f(x) = $
\item $\displaystyle \lim_{x \to 1^+} f(x) = $
\end{enumerate} 
}

\prob{
The altitude of a model rocket (in meters) $t$ seconds after launch is given by \[
h(t) = \begin{cases} 40t^2 & \text{ if } t \leq 2 \\ 160+160(t-2)-4(t-2)^2 & \text{ if } t > 2 \end{cases} 
\]
%This is a piecewise function because the rocket engine stops 2 seconds into the flight, after which the rocket moves only under the influences of gravity and friction.

\begin{enumerate}[a)]

\item Why do you think this is a piecewise function?

\vfill

\item Sketch a graph of the rocket's height for $t \geq 0$.

\noindent \begin{tikzpicture}
\draw[step=0.6cm,color=lightgray,very thin] (-4,-4) grid (4,4);
\end{tikzpicture}

\item Is $h(t)$ continuous? Should it be continuous?

\end{enumerate}
}

\end{document}