How do I align equations in three columns, justified right, center and left?

Question

\[\begin{array}{crl}
&(L-\epsilon)(x-M)\leq f(x)-f(M)&\leq(L+\epsilon)(x-M)\\ 
\Rightarrow&(L-\epsilon)(x-M)+f(M)&\leq f(x)&\leq(L+\epsilon)(x-M)+f(M)\\
\Rightarrow&(L-\epsilon)(1-\frac{M}{x})+\frac{f(M)}{x}&\leq\frac{f(x)}{x}&\leq\frac{f(M)}{x}+(L+\epsilon)(1-\frac{M}{x})
\end{array}\]

I have not been able to write in an alignment.

Answer 1

Something like this, with eqparbox?

\documentclass{article}
\usepackage{geometry}%
\usepackage{mathtools}
\usepackage{eqparbox}
\newcommand{\eqmathbox}[2][M]{\eqmakebox[#1]{$\displaystyle#2$}}

\begin{document}

\begin{spreadlines}{1.5ex}
\begin{alignat}{3}
 & & (L-\epsilon)(x-M) & \leq \eqmathbox{f(x)-f(M)} & &\leq(L+\epsilon)(x-M)\\
 & \Rightarrow\quad & (L-\epsilon)(x-M)+f(M)&\leq \eqmathbox{f(x)}& & \leq(L+\epsilon)(x-M)+f(M)\\
 & \Rightarrow & (L-\epsilon)\biggl(1-\frac{M}{x}\biggr)+\frac{f(M)}{x}&\leq\eqmathbox{\frac{f(x)}{x}} & &\leq\frac{f(M)}{x}+(L+\epsilon)\biggl(1-\frac{M}{x}\biggr)
\end{alignat}
\end{spreadlines}

\end{document} 

Answer 2

You can still use array but some adjustments are required. Disable the default arraycolsep via \setlength{\arraycolsep}{0pt} to allow default math spacing and \renewcommand{\arraystretch}{1.5} to resemble align. Finally, you need to also add \displaystyle like align.

\documentclass[a4paper]{article}
\usepackage{amsmath,array}
\begin{document}

\setlength{\arraycolsep}{0pt}
\renewcommand{\arraystretch}{1.5}
\[
\begin{array}{c >{\displaystyle{}}r<{{}} >{\displaystyle}c >{\displaystyle{}}l } 
            & (L-\epsilon)(x-M)\leq f(x)-f(M)\leq                        & (L+\epsilon)(x-M) & \\
\Rightarrow & (L-\epsilon)(x-M)+f(M)\leq                                 & f(x)              & \leq(L+\epsilon)(x-M)+f(M)\\
\Rightarrow & (L-\epsilon)\biggl(1-\frac{M}{x}\biggr)+\frac{f(M)}{x}\leq & \frac{f(x)}{x}    & \leq\frac{f(M)}{x}+(L+\epsilon)\biggl(1-\frac{M}{x}\biggr)
\end{array}
\]

\end{document}

Answer 3

Like this?

\documentclass{book}

\usepackage{mathtools}

\begin{document}

\[
\begin{array}{crcl}
    & (L-\epsilon)(x-M) \leq& f(x)-f(M)  &\leq (L+\epsilon)(x-M) \\ 
\Rightarrow & (L-\epsilon)(x-M)+f(M) \leq & f(x)&\leq(L+\epsilon)(x-M)+f(M)\\
\Rightarrow&(L-\epsilon)(1-\frac{M}{x})+\frac{f(M)}{x}\leq &\frac{f(x)}{x}&\leq\frac{f(M)}{x}+(L+\epsilon)(1-\frac{M}{x})
\end{array}\]

\end{document}