\documentclass[oneside]{amsart}

% Boilerplate to put your name on every page
% This makes grading easier
\usepackage{authoraftertitle}
\usepackage{fancyhdr}
\pagestyle{fancy}
\fancyhead{}
\fancyfoot{}
\lhead{\MyAuthor}
\rhead{\thepage}

% Use letters in enumerated lists
\usepackage{enumitem}
\setenumerate[0]{label=(\alph*)}

% You can define abbreviations here
\newcommand{\problem}[1]{\par\noindent\textbf{Problem~#1}}
\newcommand{\answer}[1]{{\par\medskip\sffamily#1\par\medskip}}
\newcommand{\cl}{\mathop{\mathrm{cl}}\nolimits}
\newcommand{\R}{\mathbb{R}}

\begin{document}

% Put your title, name, and the date here
\title{Homework 7}
\author{Your Name Here}
\date{April 1, 2020}

\maketitle

\problem{2.1.4.} The following result is called the \emph{squeezing principle}. Suppose $f$, $g$ and $h$ are functions defined on some domain $X\subseteq\mathbb{R}$ and that $a\in\cl X$. Show that if $g(x)\leq f(x)\leq h(x)$ for all $x\in X$, and if 
$$\lim_{x\to a}g(x)=\lim_{x\to a}h(x)=L,$$
then 
$$\lim_{x\to a}f(x)=L.$$

\answer{Type your answer here. Use double line breaks for paragraph breaks. 

Enclose mathematics in single dollar signs \texttt{\$f(x)\$} or in backslash-parentheses \texttt{\char`\\(f(x)\char`\\)}.

Enclose displayed mathematics in double dollar signs \texttt{\$\$f(x)\$\$} or in backslash-brackets \texttt{\char`\\[f(x)\char`\\]}.

}

\clearpage

\problem{2.1.8.} Find examples of functions \(f\) and \(g\) defined on all of \(\R\) (or on \(\R\setminus\{0\}\)) such that the following happen.

\begin{enumerate}
    \item $\lim_{x\to 0} f(x)=\infty$ and $\lim_{x\to 0} g(x)=-\infty$, but
    \[\lim_{x\to 0} f(x)+g(x)=0.\]
    
    \item $\lim_{x\to 0} f(x)=\infty$ and $\lim_{x\to 0} g(x)=-\infty$, but
    \[\lim_{x\to 0} f(x)+g(x)=-\infty.\]

    \item $\lim_{x\to 0} f(x)=\infty$ and $\lim_{x\to 0} g(x)=-\infty$, but
    \[\lim_{x\to 0} f(x)+g(x)\]
    does not exist.

    \item Is it possible for $\lim_{x\to 0} f(x)=\infty$ and $\lim_{x\to 0} g(x)=-\infty$, but
    \[\lim_{x\to 0} f(x)+g(x)=\infty?\]
\end{enumerate}

\answer{
\begin{enumerate}
    \item Answer to part (a)
    \item Answer to part (b)
    \item
    \item
\end{enumerate}
}

\clearpage

\problem{2.2.3.} Let $a<c<b$ and suppose that $f:(a,c]\to\R$ and $g:[c,b)\to\R$ are continuous functions with $f(c)=g(c)$ and such that $f$ is differentiable on $(a,c)$ and $g$ is differentiable on $(c,b)$. If we define $h:(a,b)\to\R$ by
$$h(x)=\begin{cases}f(x)&\text{if }a<x\leq c,\\g(x)&\text{if }c\leq x<b,\end{cases}$$
give a necessary and sufficient condition for $h$ to be differentiable at $x=c$.

\answer{Type answer here.}

\end{document}