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5.14 The Limit

— library procedure: limit proc x1 x2 k
— library procedure: limit proc x1 x2

Proc must be a procedure taking a single inexact real argument. K is the number of points on which proc will be called; it defaults to 8.

If x1 is finite, then Proc must be continuous on the half-open interval:

( x1 .. x1+x2 ]

And x2 should be chosen small enough so that proc is expected to be monotonic or constant on arguments between x1 and x1 + x2.

Limit computes the limit of proc as its argument approaches x1 from x1 + x2. Limit returns a real number or real infinity or ‘#f’.

If x1 is not finite, then x2 must be a finite nonzero real with the same sign as x1; in which case limit returns:

(limit (lambda (x) (proc (/ x))) 0.0 (/ x2) k)

Limit examines the magnitudes of the differences between successive values returned by proc called with a succession of numbers from x1+x2/k to 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 proc.

If the magnitudes of differences are increasing as fast or faster than a hyperbola matching at x1+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 x1+x2, then the limit is extrapolated from the quadratic passing through the three samples closest to x1.

If the magnitudes of differences are not monotonic or are not completely within one of the above categories, then #f is returned.

     ;; 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