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\chapter[MRVLIMIT: Limits of ``exp-log'' functions]%
{MRVLIMIT: Package for Computing Limits of "Exp-Log" Functions}
\label{MRVLIMIT}
\typeout{{MRVLIMIT: Package for Computing Limits of "Exp-Log" Functions}}

{\footnotesize
\begin{center}
Neil Langmead \\
Konrad-Zuse-Zentrum f\"ur Informationstechnik Berlin (ZIB) \\
Takustra\"se 7 \\
D - 14195 Berlin-Dahlem, Germany \\
\end{center}
}
\ttindex{MRVLIMIT}
%\markboth{CHAPTER \ref{MRVLIMIT}. MRVLIMIT: LIMITS OF ``EXP-LOG'' FUNCTIONS}{}
%\thispagestyle{myheadings}

Using the LIMITS package to compute the limits of functions containing
exponential and logarithmic expressions may raise a problem. For the computation
of indefinite forms (such as $0/0$,or $\frac{\infty}{\infty}$) L'Hospital's 
rule may only be applied a finite number of times in a CAS. In REDUCE it is 
applied 3 times. This algorithm of Dominik Gruntz of the ETH Z\"urich
solves this particular problem, and enables the computation of many more
limit calculations in REDUCE.


\begin{verbatim}
1: load limits;

2: limit(x^7/e^x,x,infinity);

              7
             x
      limit(----,x,infinity)
              x
             e

3: load mrvlimit;

4: mrv_limit(x^7/e^x,x,infinity);

      0
\end{verbatim}

For this example, the MRVLIMIT package is able to compute the correct limit. \\
\ttindex{MRV\_LIMIT}
\vspace{.1in}
\noindent {\tt MRV\_LIMIT}(EXPRN:{\em algebraic}, VAR:{\em kernel},
LIMPOINT:{\em algebraic}):{\em algebraic} \ttindex{MRV\_LIMIT} \par
The result is the limit of EXPRN as VAR approaches LIMPOINT. 
\vspace{.1in}

A switch {\tt TRACELIMIT} is available to inform the user about the computed
Taylor expansion, all recursive calls and the return value of the
internally called function {\tt MRV}. \\
\\
{\bf Examples}:
\\
\begin{verbatim}
5: b:=e^x*(e^(1/x-e^-x)-e^(1/x));

                -1        - x
           x + x      - e
      b:= e       *(e         - 1)


6: mrv_limit(b,x,infinity);


      -1

                                     -1
7: ex:= - log(log(log(log(x))) + log(x))  *log(x)

                       *(log(log(x)) - log(log(log(x)) + log(x)));


                  - log(x)*(log(log(x)) - log(log(log(x)) + log(x)))
      ex:=     -----------------------------------------------------
                          log(log(log(log(x))) + log(x))

8: off mcd;

9: mrv_limit(ex,x,infinity);

      1

\end{verbatim}