======= At 2005-06-14, 03:33:33 Aubrey Jaffer wrote: =======

>That bug has been corrected:
>
>> (limit (lambda (x) (* -1 x (log x))) 0 1e-9)
>102.70578127633066e-12
>> (limit (lambda (x) (* -1 x (log x))) 0 1e-222)
>0.0
>
>http://savannah.gnu.org/cgi-bin/viewcvs/scm/scm/Transcen.scm?rev=HEAD&content-type=text/vnd.viewcvs-markup
>has the new limit code.
>
>http://swiss.csail.mit.edu/~jaffer/III/Limit.html
>explains the mathematics and algorithm.
>
>The documentation for the new limit procedure is appended.
>
> | When you try to fix the reference implementation, you will find that
> | it cannot be fixed because it comes from the "black box" procedure:
>
>It took some hard work, but the new limit procedure does treat its
>procedure argument as a black box, never calling it at the limit
>point.  It essentially detects whether the function is convex or
>concave near the limit point and uses polynomial extrapolation, either
>from the function or its inverse, to estimate the limit value.  The
>inverse function extrapolation is currently limited to quadratic;
>higher order fitting can certainly be done, but the equations get big.
>
> | Bottom line: In the end, LIMIT might do more harm than it is worth.
> | You might want to reconsider if it is a feature that is essential for
> | the Scheme programming language itself.
>
>The new limit procedure works better than I had expected; even
>handling a nasty example from an old Macsyma manual (see appended
>examples).  Arguments that limits are too esoteric or new
>counterexamples may yet scuttle them from srfi-70.
>

A counterexample:

> (limit (lambda (x) (exp (expt x -2))) 0.1 0.2)
+1/0
> (exp (expt 0.1 -2))
2.6881171418161084e+043
> (limit (lambda (x) (exp (expt x -2))) 0.1 0.1000001)
+1/0
> (limit (lambda (x) (exp (expt x -2))) 0.1 0.1000000001)
+1/0

As I understand the problem, a 'limit that treats the procedure as
a "black box" is unsolvable. Even if you fix this particular
counterexample, the new 'limit will have other counterexamples.
Something similar to the halting-problem.

The additional argument k itself shows that the 'limit seems
un-Scheme-ish. Could you tell another Scheme procedrue (either
from R?RS or some other SRFI) that has a similar additional
argument?

For Computer Algebra Systems, procedures such as 'limit are only
supposed to "do right" for daily inputs. But Scheme is requested
to be "well-defined" "unambiguous, aesthetically appealing, and
consistent specications". Copying something from CASs into Scheme
is is both overly ambitious and unnecessary.

-
Chongkai Zhu
The right idea was actually had very early, but somehow didn't win for a long time.