\documentclass[12pt]{amsart}
\textwidth=16cm
\usepackage{amsfonts}
\usepackage{amsthm}
\usepackage[mathscr]{eucal}
\usepackage[all]{xy}
\usepackage{color}
\def\dsp{\displaystyle}

\begin{document}
\begin{enumerate}
\LARGE
\item [\textcolor{blue}{1.}]\textcolor{blue}{For every $\epsilon > 0 $ , there exists $\delta
> 0$ such that

\textcolor{blue}{$\mid f(x)-\it l \mid < \epsilon$ for all x with $ 0 < \mid x - a
\mid < \delta. $}

\textcolor{blue}{(or For every $\epsilon > 0 $ , there exists a corresponding
$\delta > 0$ such that for all x ,$  $ $ 0 < \mid x - a \mid <
\delta$}

\textcolor{blue}{$\Rightarrow~\mid f(x) - \it l \mid <
\epsilon.$ )}

\item [\textcolor{blue}{2.}]\textcolor{blue}{Arbitrarily give $\epsilon > 0 $. $\dsp \lim_{x \rightarrow a} f(x)= \it l
$}

\textcolor{blue}{
$\Rightarrow$ there exists $\delta_1  > 0$  such that}

\textcolor{blue}{($0 < \mid x - a \mid < \delta_1$
 $ \Rightarrow \mid f(x) - \it
l \mid < \frac{\epsilon}{2}$$) \quad \quad \quad \ldots (1)$}

\textcolor{blue}{$\dsp \lim_{x \rightarrow a} g(x)= \it m $}

\textcolor{blue}{$\Rightarrow$ there exists $\delta_2  > 0$  such
that}

\textcolor{blue}{($0 < \mid x - a \mid < \delta_2$
 $ \Rightarrow \mid g(x) - \it
m \mid < \frac{\epsilon}{2}$$) \quad \quad \quad \ldots (2)$}

\textcolor{blue}{Take $\delta = \min \{\delta_1,\delta_2\} $ , then $0 < \mid x -
a \mid < \delta $}

\textcolor{blue}{$ \Rightarrow$ $\left\{ \begin{array}{ll}
           0 < \mid x - a \mid < \delta_1 \\
           0 < \mid x - a \mid < \delta_2
        \end{array} \right.$}

\textcolor{blue}{$ \Rightarrow$ $\left\{ \begin{array}{ll}
           \mid f(x) - \it l $ $\mid < \frac{\epsilon}{2} \quad \quad ($ ${}^\cdot{}_\cdot{}^\cdot (1)$ $) \\
           \mid g(x) - \it m $$\mid < \frac{\epsilon}{2} \quad \quad ($ ${}^\cdot{}_\cdot{}^\cdot (2)$ $)
        \end{array} \right.$}

\textcolor{blue}{$\Rightarrow \mid ( f(x) + g(x) - (\it l+m ) \mid
$}

\textcolor{blue}{$=\mid ( f(x)- l ) + ( g(x) - m ) \mid$}

\textcolor{blue}{$\leq \mid f(x)- l \mid + \mid g(x) - m
\mid<\epsilon$}

\textcolor{blue}{${}_\cdot{}^\cdot{}_\cdot$ For an arbitrarily given $\epsilon > 0
$ , there exist $\delta > 0 $ such that $ 0 < \mid x - a \mid <
\delta_1 $}

\textcolor{blue}{$\Rightarrow \mid f(x) + g(x) - (\it l + m )
\mid < \epsilon. $}

\textcolor{blue}{Hence $\dsp \lim_{x\rightarrow a} \it f(x) + g(x) = l + m
$.}

\end{enumerate}

\end{document}
