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| Date: Wed, 06 Jul 2005 13:55:40 -0400 | From: Paul Schlie <schlie@xxxxxxxxxxx> | | 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.