| Date: Wed, 06 Jul 2005 13:55:40 -0400
 | From: Paul Schlie <xxxxxx@comcast.net>
 |
 | As one last thought:
 |
 | If the default value of a function were defined as the average value of
 | it's limits, then it may be reasonable to define a number system like:
 |
 |  -1.0 -10. -1/0   +1/0 +10. +1.0      -1.0 -10. -Inf   Inf 10. 1.0
 |  -------------- 0 --------------  ::  -------------- 0 -----------
 |  -1.0 -0.1 -0/1   +0/1 +0.1 +1.0      -1.0 -0.1 -0.0   0.0 0.1 1.0
 |
 | Where absolute zero is designated as 0, and who's reciprocal is 0, as
 | the average value of it's -1/0 and +1/0 limits would be 0; as would 0/0,

Then the FINITE? predicate becomes useless.

 | and the difference of any two equivalent values, thereby eliminating the
 | otherwise complexity and arguably negligible value of an inexact 0. i.e.:
 |
 | (= -0.0 0 +0.0) => #t, (< -0.0 0 +0.0) => #t, and (< -1/0 0 +1/0) => #t
 |
 | Thereby all functions will be legitimately valued at all points with no need
 | of ambiguous value representation, however who's value may be more precisely
 | determined at a specific limit through the use of a limit macro as desired.

Why do you feel compelled to turn LIMIT into a macro?

 | Thereby hypothetically: (presuming sufficient numerical precision)
 |
 | (tan pi/2) => 0

An exact zero?  That is just wrong.

 | (limit (lambda (x) (tan x)) (pi/2 -0/1)) => +1/0
 | (limit (lambda (x) (tan x)) (pi/2 +0/1)) => -1/0
 |
 | (+ 4. (/ (abs 0) 0)) => 4.0
 | (limit (lambda (x) (+ 4. (/ (abs x) x))) (0 -0/1)) => 3.0
 | (limit (lambda (x) (+ 4. (/ (abs x) x))) (0 +0/1)) => 5.0

LIMIT already handles these cases correctly.  But I am unconvinced
that a procedure can automatically pick the evaluation points given no
information about the test function.