\documentclass[12pt]{amsart}

% Package Being Used:

\usepackage{enumitem}
\usepackage{bigints}
\usepackage{amssymb}
\usepackage{bm}
\usepackage{graphicx}
\usepackage{psfrag}
\usepackage{color}
\usepackage{url}
\usepackage{algpseudocode}
\usepackage{fancyhdr}
\usepackage{xy}
\input xy
\xyoption{all}

%\usepackage{soul} % this is a package used only temporary that allows us to underline comments that we are planning to remove in future versions of the paper.

% Paper Format and Geometry:

\voffset=-1.4mm
\oddsidemargin=17pt
\evensidemargin=17pt
\topmargin=26pt
\headheight=9pt     
\textheight=576pt
\textwidth=440.8pt
\parskip=0pt plus 4pt

% Head Labels:

\pagestyle{fancy}
\fancyhf{}
\renewcommand{\headrulewidth}{0pt}
\renewcommand{\footrulewidth}{0pt}
%\fancyhead[CE]{\fontsize{9}{11}\selectfont F. GOTTI AND M. GOTTI}
%\fancyhead[CO]{\fontsize{9}{11}\selectfont ATOMICITY AND BOUNDEDNESS OF MONOTONE PUISEUX MONOIDS}
%\fancyhead[LE,RO]{\thepage}

% Theorems-like Format and Numbering:


\newtheorem{theorem}{Theorem}%[section]
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{problem}[theorem]{Problem}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{cor}[theorem]{Corollary}
\newtheorem{conjecture}[theorem]{Conjecture}
\theoremstyle{definition}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{example}[theorem]{Example}
\numberwithin{equation}{section}

% Personalized Commands:

\newcommand\cc{\mathbb{C}}
\newcommand\ff{\mathbb{F}}
\newcommand\nn{\mathbb{N}}
\newcommand\qq{\mathbb{Q}}
\newcommand\rr{\mathbb{R}}
\newcommand\zz{\mathbb{Z}}

\newcommand\pval{\mathsf{v}_p}

\newcommand{\twopf}[4]
{
	\left\{
	\begin{array}{ll}
		#1 & \mbox{if } #2 \\
		#3 & \mbox{if } #4
	\end{array}
	\right.
}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{document}
	
	\mbox{}
	\title{Problem Set 4: Precise Concepts of Limit and Continuity}
	
	\maketitle
	
	
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%
	%\section{Introduction} \label{sec:intro}
	\noindent {\footnotesize Note: Most of the problems were taken from the textbook} \cite{jS15}.
	
%	\begin{problem}
%		Find the equation of a tangent line to the graph of the function $f(x) = x^2 + 4x + 3$ passing by the point $P = (0,-2)$.
%	\end{problem}

	\begin{problem}
		Prove the following statements using the $\epsilon-\delta$ definition of limit.
		\begin{enumerate}[label=\alph*), topsep=8pt, itemsep=8pt, partopsep=1ex, parsep=1ex]
			\item $\lim_{x \to 1} \frac{2 + 4x}3 = 2$;
			\item $\lim_{x \to a} x = a$;
			\item $\lim_{x \to 0} x^3 = 0$;
			\item $\lim_{x \to 2} (x^2 - 4x + 5) = 1$;
			\item $\lim_{x \to a} \sqrt{x} = \sqrt{a}$.
		\end{enumerate}
	\end{problem}

	\begin{problem}
		If the function $f$ is defined by
		\[
			f(x) = \twopf{0}{\ x \in \qq}{1}{\ x \in \rr \setminus \qq},
		\]
		then prove that $\lim_{x \to 0} f(x)$ does not exist.
	\end{problem}

	\begin{problem}
		Suppose that $\lim_{x \to a} f(x) = \infty$ and $\lim_{x \to a} g(x) = c$, where $c > 0$. Prove that
		\[
			\lim_{x \to a} [f(x) + g(x)] = \infty \quad \text{and} \quad \lim_{x \to a} [f(x)g(x)] = \infty.
		\]
	\end{problem}

	\begin{problem}
		Why are the following functions continuous at every point of its domain. State the domain.
		\begin{enumerate}[label=\alph*), topsep=8pt, itemsep=8pt, partopsep=1ex, parsep=1ex]
			\item $F(x) = \sin(\cos(\sin x))$;
			\item $B(x) = \frac{\tan x}{\sqrt{4 - x^2}}$;
			\item $N(x) = \tan^{-1}(1 + e^{-x^2})$.
		\end{enumerate}
	\end{problem}

	\begin{problem}
		Use continuity to evaluate $\lim_{x \to 2} x \sqrt{20 - x^2}$ and $\lim_{x \to 4} 3^{\sqrt{x^2 - 2x -4}}$.
	\end{problem}

	\begin{problem}
		Sketch the graph of the following function:
		\[
			f(x) = \twopf{1-x^2}{x < 1}{1/x}{x \ge 1}.
		\]
		Is the function $f$ continuous everywhere? Explain.
	\end{problem}

	\begin{problem}
		If $f(x) = x^2 + 10 \sin x$, show that there is a number $c$ such that $f(c) = 1000$.
	\end{problem}

	\begin{problem}
		Use the Intermediate Value Theorem to show that there is a root of the given equation in the specified interval.
		\begin{enumerate}[label=\alph*), topsep=8pt, itemsep=8pt, partopsep=1ex, parsep=1ex]
			\item $x^2 + x - 3 = 0$, \quad $(1,2)$;
			\item $\sqrt[3]{x} = 1 - x$, \quad $(0,1)$;
			\item $\sin x = x^2 - x$, \quad $(1,2)$.
		\end{enumerate}
	\end{problem}

	\begin{problem}
		Prove that each of the following equations has at least one real root.
		\begin{enumerate}[label=\alph*), topsep=8pt, itemsep=8pt, partopsep=1ex, parsep=1ex]
			\item $\cos x = x^3$;
			\item $x^5 - x^2 + 2x + 3 = 0$;
			\item $x^5 - x^2 - 4 =0$.
		\end{enumerate}
	\end{problem}
	
	\vspace{20pt}
	
	\begin{thebibliography}{20}
		
		\bibitem{jS15} J.~Stewart: \emph{Single Variable Calculus} 8th Edition, Cengage Learning, Boston 2015.
		
	\end{thebibliography}
	
\end{document}