\documentclass[12pt]{amsart}
\usepackage{amsfonts}
\usepackage{amsthm}
\usepackage[mathscr]{eucal}
\usepackage[all]{xy}
\usepackage{color}
\def\dsp{\displaystyle}


\title{Limit and Continuity}

\begin{document}

\maketitle
\begin{enumerate}
\item[]非管理學院 (理,工,資電,地科學院)

\quad

\item[Problems:]

\item [1.]Give the $\epsilon - \delta$ definition of $\dsp \lim_{x
\rightarrow a} f(x) = \it{l}$.

\


\item [2.]Use $\epsilon - \delta$ definition to prove the following :
$$
 if \quad \dsp \lim_{x \rightarrow a} f(x) = \it{l} ,\quad \dsp \lim_{x \rightarrow a} g(x) = \it{m}
$$
$$
then\quad \dsp \lim_{x \rightarrow a} f(x)+g(x) = \it{l} + m
$$

\

\item [3.]State the Sandwich Theorem. (i.e. Squeezing Theorem )

\

\item [4.]Give the definition of "f is continuous at a".

\

\item [5.]f(x)=$
\left\{ \begin{array}{ll}
           x^2 + 2  \quad \quad   x<1 \\
           \frac{1}{x + 1} \quad \quad \quad x>1
        \end{array} \right.$
\item[] Write down \quad (1) $\dsp \lim_{x\rightarrow 1^-} f(x)$ \quad (2) $\dsp \lim_{x\rightarrow 1^+} f(x)$ \quad (3) $\dsp
\lim_{x\rightarrow 1} f(x) \quad $

\

\item [6.]g(x)=$ \left\{ \begin{array}{ll}
           x^3 + 1 \\
           3 \\
           x + 2
        \end{array} \right.$
        $\begin{array}{clcr}
x < 0   \\
x = 0  \\
x > 0
\end{array}$
\item[] (i)Write down \quad (1) $\dsp \lim_{x\rightarrow 0^-} g(x)$ \quad (2) $\dsp \lim_{x\rightarrow 0^+} g(x)$ \quad (3) $\dsp
\lim_{x\rightarrow 0} g(x)  $
\item[] (ii)Is g continuous at 0 ? Why ?

\

\item [7.]h(x)=$
\left\{ \begin{array}{ll}
           x^2 + 1 \\
           2 \\
           -2x
        \end{array} \right.$
        $\begin{array}{clcr}
x < -1   \\
x = \quad  1  \\
x > -1
\end{array}$
\item[] (i)Write down \quad (1) $\dsp \lim_{x\rightarrow -1^-} h(x)$ \quad (2) $\dsp \lim_{x\rightarrow -1^+} h(x)$ \quad (3) $\dsp
\lim_{x\rightarrow -1} h(x)  $
\item[] (ii)Is h continuous at -1 ? Why ?

\

\item [8.]f(x)=$
\left\{ \begin{array}{ll}
           x \\
           -x
        \end{array} \right.$
        $\begin{array}{clcr}
x \quad  is \quad   rational  \\
x \quad  is \quad  irrational
\end{array}$

\

\item[] (i)Write down \quad (1) $\dsp \lim_{x\rightarrow 0} f(x)$ \quad (2) $\dsp \lim_{x\rightarrow 1} f(x)$
\item[] (ii)Write down the points at which f is continuous.
\

\item [9.]Write down $\dsp \lim_{x\rightarrow 0} \sin \frac{1}{x}$ , and justify your
answer.

\

\item [10.]Write down $\dsp \lim_{x\rightarrow 0} x\sin \frac{1}{x}$ , and justify your
answer.

\

\item [11.]Give the definition of $\dsp \lim_{x\rightarrow a} f(x)=\infty$

\end{enumerate}

\end{document}