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\title{Math 120 Homework 2}
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\begin{itemize}
\item Due Friday September 22 at start of class.
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\item If your homework is longer than one page, {\bf staple} the pages together, and put your name on each sheet of paper. {\bf Homework lacking a staple will lose 2 points}.
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%\item {\bf Collaboration Policy}: You are welcome (and encouraged) to work on the homework in groups. However, each student must write up the homework on their own, and must use their own wording (i.e.~don't jusy copy the solutions from your friend). If you do collaborate with others, please list the name of your collaborators at the top of the homework.

%\item You are encouraged (though not required) to type up your solutions. If you choose to do this, I strongly recommend that you use the typesetting software LaTeX. LaTeX is used by the entire mathematics community, and if you intend to go into math, you’ll need to learn it sooner or later. ``The Not So Short Introduction to LaTeX'' is a good place to start. This guide can be found at http://tug.ctan.org/info/lshort/english/lshort.pdf . You can also download the .tex source file for this homework and take a look at that.

\item Each homework problem should be correct as stated. Occasionally, however, I might screw something up and give you an impossible homework problem. If you believe a problem is incorrect, please email me. If you are right, the first person to point out an error will get +1 on that homework, and I will post an updated version. 
\end{itemize}

\subsection*{Functions}
 {\bf 1.} Using the domain convention, state the domain and range of each of the following functions. You do not need to prove that your answer is correct.\\
 \noindent a) $f(x) = x^2 - 1$\\
 \noindent b) $f(x) = \cos(1/x)$\\
 \noindent c) $f(x) = x^3 - \cos(1/x)$\\

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\noindent {\bf 2}. Let $f(x) = 1/x + 2$, $g(x) = \sin(x)$. What is the domain of $f\circ g$? You do not need to prove that your answer is correct.

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\subsection*{Universal Quantifiers}

\noindent {\bf 3}. Write down the negation of each of the following expressions. You do not need to determine whether the expressions are true or false\\
a) $\forall\ x \in \RR,\ \exists\ y \in \RR\ \st x^2 > y$.\\
b) $\forall\ x \in \RR\ \textrm{with}\ x>0,\ \exists\ y\in\RR\ \st x > y^2$.\\
c) $\forall\ x \in \RR\ \textrm{with}\ x>0,\ \exists\ y\in\RR\ \st \forall\ z\in\RR,\textrm{ we have}\ x > y^2 - z$.\\
d) $\forall\ \epsilon \in\RR\ \textrm{with}\ \epsilon>0,\ \exists\ \delta\in\RR\ \textrm{with}\ \delta>0\ \st \forall\ x\in\RR\ \textrm{with}\ |x-3|<\delta,\textrm{ we have}\ |x^2 - 8|<\epsilon$.

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\noindent {\bf 4.} For each of the expressions from Question 3, determine whether the expression is true, or its negation is true, and (briefly) explain why.


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\subsection*{Limits}
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\noindent {\bf 5.} Using the definition of a limit, prove that the following two statements are true.\\
a) $\lim_{x\to 2} x^2 + x = 6.$\\
b) $\lim_{x\to 1} x^3 + x^2 + x  = 3$. \\
Hint: don't forget to verify the domain requirement!

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\noindent {\bf 6.} Using the definition of a limit, prove that the following two statements are false.\\
a) $\lim_{x\to 10} 3x = 29.$\\
b) $\lim_{x\to 1}(x^2 / x) = 2$.

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\noindent  {\bf 7.} Let $f(x) = x,\ g(x) = x$.\\
 a) What is the domain of $(f/g)(x)$?.\\
 b) Using the definition of a limit, prove that $\lim_{x\to 0}(f/g)(x) = 1$.


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