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\noindent
\parbox{2.25in}{\bf Math 3220}
\hfill {\Large \bf Homework \#5} \hfill
\parbox{2.25in}{\bf \hfill Due: Fri., Mar. $15^\mathrm{th}$, 2019}

\vspace{0.2in}

%\noindent {\bf\large Name:} \underline{\qquad\sc\large This is my fabulous name! \qquad}
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\begin{description}
\item[\#1 Defining Terms] State the definition of $\lim\limits_{x \to a^{+}} f(x) = -\infty$. 

Adapt one of the ``variant limit definitions'' given in class. Make sure you include an appropriate assumption about the domain of $f$.

\item[\#2 Take it to the Limit] Use the $\epsilon$-$\delta$ definition of the limit (or one-sided limit) to prove the following:

\begin{enumerate}[(a)]
\item $\lim\limits_{x \to 2} x^2 = 4$

\item $\lim\limits_{x \to 1^-} \sqrt{1-x\,\strut}=0$
\end{enumerate}

\item[\#3 One More Time] Let $f(x) = \left\{ \begin{array}{ccc} \strut  3x+1 & & x \geq 1 \\ 2 & & x < 1 \end{array} \right.$ 

Obviously, $\lim\limits_{x \to 1} f(x)$ does not exist.

\begin{enumerate}[(a)]
\item Use (the negation of) the $\epsilon$-$\delta$ definition of the limit to show that this limit does not exist.

\item Give a quick argument that this limit does not exist using left and right handed limits.
\end{enumerate}

\item[\#4 Squeezing In Another Limit] Let $f$, $g$, and $h$ be defined on $I-\{a\}$ where $I$ is an open interval containing $a$. In addition, suppose that $f(x) \leq g(x) \leq h(x)$ for all $x \in I -\{a\}$. Finally, assume that $\lim\limits_{x \to a} f(x) = \lim\limits_{x \to a} h(x) = L$. Prove that $\lim\limits_{x \to a} g(x) = L$.

Prove this using Lemma 3.2 (i.e., use the squeeze theorem for sequences to get this squeeze theorem).

\item[\#5 Linear Things are Continuous] Let $A,B \in \mathbb{R}$ and define $f:\mathbb{R} \to \mathbb{R}$ by $f(x)=Ax+B$. Prove that $f$ is continuous (everywhere). Do this using the $\epsilon$-$\delta$ definition of continuity.

\item[\#6 Continuity Continued] Let $f: \mathbb{R} \to \mathbb{R}$ be a continuous function. Suppose that $f(r)=0$ for all $r \in \mathbb{Q}$ ($f$ is zero on all of the rational numbers). Show that $f(x)=0$ for all $x \in \mathbb{R}$.

\item[RESUBMIT] Type up Homework \#4 Problem \#3 and its solution in \LaTeX. 

\end{description}

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