\documentclass[12pt,fleqn]{article}
\pagestyle{empty}
\setlength{\topmargin}{-1in}
\setlength{\oddsidemargin}{-.3in}
\setlength{\textheight}{9in}
\setlength{\textwidth}{7.2in}

\begin{document}

\begin{center}
{\Large TEST 1

Math 111}
\end{center}
\vspace{.1in}
You have 1 hour to complete the test.  You may not use a calculator.
Show all relevant work (or else no credit).  
Use the backs of these pages if necessary.

\vspace{.1in}

Name \rule{5in}{.2pt}
\vspace{.1in}
\begin{enumerate}

\item  (10 points)
\begin{enumerate}
\item  What does a function have to satisfy to be continuous at a point?  
\vspace{1.5in}
\item  What does a function have to satisfy to be continuous?
\vspace{1.5in}
\end{enumerate}

\item  (10 points)  True or False, circle the correct answer:

\begin{tabular}{llll}
\\
T & F &  & For all functions $f$ and $g$, $f\circ g=g\circ f$. \\
\\
\\
T & F &  & For all functions, $f$, $\lim_{x\rightarrow c}f(x)=f(c)$ \\
\\
\end{tabular}
\newpage
%\vspace{.5in}

\item  (10 points)  In the equation below, 
can $y$ be expressed as a function of $x$? ($y=f(x)$)
If it can be done, find the function.  If it cannot, why not?
\[
\sin x+2y+xy+x^2y=3
\]
\vspace{2.5in}
%\newpage

\item  (10 points)  Find $f$ and $g$ so that $h=f\circ g$.  
(Make sure that neither $f(x)=x$ nor $g(x)=x$)
\[
h(x)=\sin(x^2+\cos x)
\]
\vspace{2.4in}

\item (10 points)  Find the limit
\[
\lim_{x\rightarrow 4} \frac{x-4}{\sqrt{x}-2}
\]
%\vspace{2in}
\newpage

\item (12 points)  
Find the limits or state that the limit does not exist for 
the function $g$ graphed below.
\begin{enumerate}
\item  \[ \lim_{x\rightarrow -2}g(x) \]
\vspace{.2in}
\item  \[ \lim_{x\rightarrow 1^-}g(x) \]
\vspace{.2in}
\item  \[ \lim_{x\rightarrow 1^+}g(x) \]
\vspace{.2in}
\item  \[ \lim_{x\rightarrow 1}g(x) \]
\vspace{.2in}
\item  True or False:  $g$ is continuous at 0.
\vspace{.2in}
\item  True or False:  $g$ is continuous at -2.
\end{enumerate}

\setlength{\unitlength}{3pt}
\begin{picture}(100,100)
\put(0,50){\vector(1,0){100}}
\put(50,0){\vector(0,1){100}}
\put(95,53){$x$}
\put(53,95){$y$}
\put(20,49){\line(0,1){2}}
\put(19,46){-2}
\put(35,49){\line(0,1){2}}
\put(34,46){-1}
\put(65,49){\line(0,1){2}}
\put(64,46){1}
\put(80,49){\line(0,1){2}}
\put(79,46){2}

\put(49,20){\line(1,0){2}}
\put(45,19){-2}
\put(49,35){\line(1,0){2}}
\put(45,34){-1}
\put(49,65){\line(1,0){2}}
\put(45,64){1}
\put(49,80){\line(1,0){2}}
\put(45,79){2}
\end{picture}
\newpage

\item  (9 points)  Find the following limits: (Note that $\lceil x \rceil$ is the greatest integer function)
\begin{enumerate}
\item  \[ \lim_{t\rightarrow \frac{1}{2}^+} \lceil 2t \rceil \]
\vspace{2.5in}

\item  \[ \lim_{t\rightarrow \frac{1}{2}^-} \lceil 2t \rceil \]
\vspace{2.5in}

\item  \[ \lim_{t\rightarrow \frac{1}{2}} \lceil 2t \rceil \]
%\vspace{2.5in}

\end{enumerate}
\newpage

\item  (10 points)
\begin{enumerate}
\item \label{part a} What is the natural domain of $f$?
\[
f(x)=\frac{x^3+8}{x+2}
\]
\vspace{1.3in}
\item  Can you define $f$ ($f$ as in part \ref{part a}) at the points not in the domain of $f$, so that $f$ is continuous everywhere?  If so how, if not why not?
\end{enumerate}
\vspace{2in}
%\newpage

\item  (10 points)  Given the function $f$ below, 
what do $a$ and $b$ need to be so that $f$ is continuous?
\[
f(x)=\left\{ \begin{array}{ll}
x+1  & \mbox{ if } x \leq 2 \\
ax+b & \mbox{ if } 2 < x < 3 \\
3x+1  & \mbox{ if } x \geq 3 \\
\end{array}
\right.
\]
%\vspace{3in}
\newpage
\item  (10 points)  
\[
g(x)=\sqrt{2x-1}
\]
\begin{enumerate}
\item  Using the limit definition of derivative, find $g'(x)$.
\vspace{3.4in}
\item  Find the equation of the tangent line at $x=5$ for $g$.

\end{enumerate}



\end{enumerate}



\end{document}






