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\chapter{LIMITS: A package for finding limits}
\label{LIMITS}
\typeout{{LIMITS: A package for finding limits}}

{\footnotesize
\begin{center}
Stanley L. Kameny \\
Los Angeles, U.S.A.
\end{center}
}
\ttindex{LIMITS}

LIMITS is a fast limit package for \REDUCE\ for functions which are
continuous except for computable poles and singularities, based on some
earlier work by Ian Cohen and John P. Fitch.
The Truncated Power Series
package is used for non-critical points, at which the value of the
function is the constant term in the expansion around that point.
\index{l'H\^opital's rule}
l'H\^opital's rule is used in critical cases, with preprocessing of
$\infty - \infty$ forms and reformatting of product forms in order
to apply l'H\^opital's rule.  A limited amount of bounded arithmetic
is also employed where applicable.

\section{Normal entry points}
\ttindex{LIMIT}
\vspace{.1in}
\noindent {\tt LIMIT}(EXPRN:{\em algebraic}, VAR:{\em kernel},
LIMPOINT:{\em algebraic}):{\em algebraic}
\vspace{.1in}

This is the standard way of calling limit, applying all of the
methods. The result is the limit of EXPRN as VAR approaches LIMPOINT.


\section{Direction-dependent limits}

\ttindex{LIMIT+}\ttindex{LIMIT-}
\vspace{.1in}
\noindent {\tt LIMIT!+}(EXPRN:{\em algebraic}, VAR:{\em kernel},
LIMPOINT:{\em algebraic}):{\em algebraic} \\
\noindent {\tt LIMIT!-}(EXPRN:{\em algebraic}, VAR:{\em kernel},
LIMPOINT:{\em algebraic}):{\em algebraic}
\vspace{.1in}

If the limit depends upon the direction of approach to the {\tt
LIMPOINT}, the functions {\tt LIMIT!+} and {\tt LIMIT!-} may be used.
They are defined by:

\vspace{.1in}
\noindent{\tt LIMIT!+} (EXP,VAR,LIMPOINT) $\rightarrow$
\hspace*{2em}{\tt LIMIT}(EXP*,$\epsilon$,0) \\
where EXP* = sub(VAR=VAR+$\epsilon^2$,EXP)

and

\noindent{\tt LIMIT!-} (EXP,VAR,LIMPOINT) $\rightarrow$
\hspace*{2em}{\tt LIMIT}(EXP*,$\epsilon$,0) \\
where EXP* = sub(VAR=VAR-$\epsilon^2$,EXP)

Examples:
\begin{verbatim}
load_package misc;

limit(sin(x)/x,x,0);

1

limit((a^x-b^x)/x,x,0);

log(a) - log(b)

limit(x/(e**x-1), x, 0);

1

limit!-(sin x/cos x,x,pi/2);

infinity

limit!+(sin x/cos x,x,pi/2);

 - infinity

limit(x^log(1/x),x,infinity);

0

limit((x^(1/5) + 3*x^(1/4))^2/(7*(sqrt(x + 9) - 3 - x/6))^(1/5),x,0);

     3/5
  - 6
---------
   1/5
  7

\end{verbatim}