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diff --git a/limit.texi b/limit.texi new file mode 100644 index 0000000..0a3a64c --- /dev/null +++ b/limit.texi @@ -0,0 +1,101 @@ +@settitle The limit procedure + +@deffn {library procedure} limit proc x1 x2 k +@deffnx {library procedure} limit proc x1 x2 + +@var{Proc} must be a procedure taking a single inexact real argument. +@var{K} is the number of points on which @var{proc} will be called; it +defaults to 8. + +If @var{x1} is finite, then @var{Proc} must be continuous on the +half-open interval: + + ( @var{x1} .. @var{x1}+@var{x2} ] + +And @var{x2} should be chosen small enough so that @var{proc} is +expected to be monotonic or constant on arguments between @var{x1} and +@var{x1} + @var{x2}. + +@code{Limit} computes the limit of @var{proc} as its argument +approaches @var{x1} from @var{x1} + @var{x2}. +@code{Limit} returns a real number or real infinity or @samp{#f}. + +If @var{x1} is not finite, then @var{x2} must be a finite nonzero real +with the same sign as @var{x1}; in which case @code{limit} returns: + +@code{(limit (lambda (x) (proc (/ x))) 0.0 (/ @var{x2}) @var{k})} + +@code{Limit} examines the magnitudes of the differences between +successive values returned by @var{proc} called with a succession of +numbers from @var{x1}+@var{x2}/@var{k} to @var{x1}. + +If the magnitudes of differences are monotonically decreasing, then +then the limit is extrapolated from the degree n polynomial passing +through the samples returned by @var{proc}. + +If the magnitudes of differences are increasing as fast or faster than +a hyperbola matching at @var{x1}+@var{x2}, then a real infinity with +sign the same as the differences is returned. + +If the magnitudes of differences are increasing more slowly than the +hyperbola matching at @var{x1}+@var{x2}, then the limit is +extrapolated from the quadratic passing through the three samples +closest to @var{x1}. + +If the magnitudes of differences are not monotonic or are not +completely within one of the above categories, then #f is returned. +@end deffn + +@example +;; constant +(limit (lambda (x) (/ x x)) 0 1.0e-9) ==> 1.0 +(limit (lambda (x) (expt 0 x)) 0 1.0e-9) ==> 0.0 +(limit (lambda (x) (expt 0 x)) 0 -1.0e-9) ==> +inf.0 +;; linear +(limit + 0 976.5625e-6) ==> 0.0 +(limit - 0 976.5625e-6) ==> 0.0 +;; vertical point of inflection +(limit sqrt 0 1.0e-18) ==> 0.0 +(limit (lambda (x) (* x (log x))) 0 1.0e-9) ==> -102.70578127633066e-12 +(limit (lambda (x) (/ x (log x))) 0 1.0e-9) ==> 96.12123142321669e-15 +;; limits tending to infinity +(limit + +inf.0 1.0e9) ==> +inf.0 +(limit + -inf.0 -1.0e9) ==> -inf.0 +(limit / 0 1.0e-9) ==> +inf.0 +(limit / 0 -1.0e-9) ==> -inf.0 +(limit (lambda (x) (/ (log x) x)) 0 1.0e-9) ==> -inf.0 +(limit (lambda (x) (/ (magnitude (log x)) x)) 0 -1.0e-9) + ==> -inf.0 +;; limit doesn't exist +(limit sin +inf.0 1.0e9) ==> #f +(limit (lambda (x) (sin (/ x))) 0 1.0e-9) ==> #f +(limit (lambda (x) (sin (/ x))) 0 -1.0e-9) ==> #f +(limit (lambda (x) (/ (log x) x)) 0 -1.0e-9) ==> #f +;; conditionally convergent - return #f +(limit (lambda (x) (/ (sin x) x)) +inf.0 1.0e222) + ==> #f +;; asymptotes +(limit / -inf.0 -1.0e222) ==> 0.0 +(limit / +inf.0 1.0e222) ==> 0.0 +(limit (lambda (x) (expt x x)) 0 1.0e-18) ==> 1.0 +(limit (lambda (x) (sin (/ x))) +inf.0 1.0e222) ==> 0.0 +(limit (lambda (x) (/ (+ (exp (/ x)) 1))) 0 1.0e-9) + ==> 0.0 +(limit (lambda (x) (/ (+ (exp (/ x)) 1))) 0 -1.0e-9) + ==> 1.0 +(limit (lambda (x) (real-part (expt (tan x) (cos x)))) (/ pi 2) 1.0e-9) + ==> 1.0 +;; This example from the 1979 Macsyma manual grows so rapidly +;; that x2 must be less than 41. It correctly returns e^2. +(limit (lambda (x) (expt (+ x (exp x) (exp (* 2 x))) (/ x))) +inf.0 40) + ==> 7.3890560989306504 +;; LIMIT can calculate the proper answer when evaluation +;; of the function at the limit point does not: +(tan (atan +inf.0)) ==> 16.331778728383844e15 +(limit tan (atan +inf.0) -1.0e-15) ==> +inf.0 +(tan (atan +inf.0)) ==> 16.331778728383844e15 +(limit tan (atan +inf.0) 1.0e-15) ==> -inf.0 +((lambda (x) (expt (exp (/ -1 x)) x)) 0) ==> 1.0 +(limit (lambda (x) (expt (exp (/ -1 x)) x)) 0 1.0e-9) + ==> 0.0 +@end example |