\documentclass{article}
\newcommand{\polim}{\lim_{\theta \rightarrow 0}}
\title{Limits by Equation!}
\author{Po-Shen Loh}
\date{11 March 2001}

\begin{document}

\maketitle

Consider the following limit; we shall solve it without resorting to
L'Hopital's rule or power series.

\begin{displaymath}
  \polim \frac{\theta-\sin \theta}{\theta^3}
\end{displaymath}

To solve this limit, we use equations:

\begin{eqnarray*}
   X &=& \lim_{\theta \rightarrow 0} \frac{\theta-\sin
   \theta}{\theta^3} \\
   &=& \polim \frac{2 (\theta/2) - 2\sin (\theta/2) \cos (\theta/2)}{\theta^3} \\
   &=& \polim \frac{2 (\theta/2) - 2\sin (\theta/2) (1 - 2 \sin^2
   (\theta/4))}{\theta^3} \\
   &=& \polim \frac{2 (\theta/2) - 2\sin (\theta/2)}{\theta^3} +
   \polim \frac{4
   \sin (\theta/2) \sin^2 (\theta/4)}{\theta^3} \\
   &=& X/4 + 1/8 \\
   \frac{3}{4} X &=& \frac{1}{8} \\
   X &=& 1/6.
\end{eqnarray*}

And we are done.

\end{document}