Thu Feb 25 19:42:28 MST 1999 aquarius% maple |\^/| Maple V Release 5.1 (WMI Campus Wide License) ._|\| |/|_. Copyright (c) 1981-1998 by Waterloo Maple Inc. All rights \ MAPLE / reserved. Maple and Maple V are registered trademarks of <____ ____> Waterloo Maple Inc. | Type ? for help. # ----------[ M a p l e ]---------- #interface(echo = 3); # ---------- Initialization ---------- > readlib(showtime): > on; # ---------- Limits ---------- # Start with a famous example => e O1 := Limit((1 + 1/n)^n, n = infinity); n lim (1 + 1/n) n -> infinity time = 0.03, bytes = 34006 O2 := value(%); exp(1) time = 0.23, bytes = 547478 # => 1/2 O3 := limit((1 - cos(x))/x^2, x = 0); 1/2 time = 0.07, bytes = 84762 # See Dominik Gruntz, _On Computing Limits in a Symbolic Manipulation System_, # Ph.D. dissertation, Swiss Federal Institute of Technology, Zurich, # Switzerland, 1996. => 5 O4 := limit((3^x + 5^x)^(1/x), x = infinity); bytes used=1000392, alloc=851812, time=0.66 bytes used=2000620, alloc=1507052, time=1.83 5 time = 1.45, bytes = 1184474 # => 1 O5 := limit(log(x)/(log(x) + sin(x)), x = infinity); infinity time = 0.10, bytes = 91062 # => - e^2 [Gruntz] O6 := Limit((exp(x*exp(-x)/(exp(-x) + exp(-2*x^2/(x + 1)))) - exp(x))/x, O6 := x = infinity); x exp(-x) exp(-----------------------) - exp(x) 2 x exp(-x) + exp(-2 -----) x + 1 lim ------------------------------------- x -> infinity x time = 0.01, bytes = 14406 O7 := value(%); -exp(2) time = 0.28, bytes = 235314 # => 1/3 [Gruntz] O8 := Limit(x*log(x)*log(x*exp(x) - x^2)^2/log(log(x^2 + 2*exp(exp(3*x^3*log(x))))), O8 := x = infinity); 2 2 x ln(x) ln(x exp(x) - x ) lim ----------------------------------- x -> infinity 2 3 ln(ln(x + 2 exp(exp(3 x ln(x))))) time = 0.02, bytes = 14214 O9 := value(%); bytes used=3001148, alloc=1769148, time=3.21 1/3 time = 1.02, bytes = 649030 # => 1/e [Knopp, p. 73] O10 := limit(1/n * n!^(1/n), n = infinity); exp(-1) time = 0.57, bytes = 443710 # Rewrite the above problem slightly => 1/e O11 := limit(1/n * gamma(n + 1)^(1/n), n = infinity); (1/n) gamma(n + 1) lim ----------------- n -> infinity n time = 0.21, bytes = 176714 # => 1 [Gradshteyn and Ryzhik 8.328(2)] O12 := limit(gamma(z + a)/gamma(z)*exp(-a*log(z)), z = infinity); gamma(z + a) exp(-a ln(z)) lim -------------------------- z -> infinity gamma(z) time = 0.24, bytes = 210994 # => e^z [Gradshteyn and Ryzhik 9.121(8)] O13 := limit(hypergeom([1, k], [1], z/k), k = infinity); lim hypergeom([k], [], z/k) k -> infinity time = 0.05, bytes = 32946 O14 := simplify(%); bytes used=4001364, alloc=1834672, time=4.64 exp(z) time = 0.38, bytes = 147298 # => Euler's_constant [Gradshteyn and Ryzhik 9.536] O15 := limit(Zeta(x) - 1/(x - 1), x = 1); gamma time = 0.07, bytes = 64790 # => gamma(x) [Knopp, p. 385] O16 := Limit(n^x/(x * Product((1 + x/k), k = 1..n)), n = infinity); x n lim ------------------------ n -> infinity / n \ | --------' | |' | | | x | | | (1 + x/k)| | | | | | | | | \ k = 1 / time = 0.00, bytes = 7210 O17 := value(%); x n GAMMA(n + 1) GAMMA(x + 1) lim ---------------------------- n -> infinity x GAMMA(n + 1 + x) time = 0.83, bytes = 644706 # See Angus E. Taylor and W. Robert Mann, _Advanced Calculus_, Second Edition, # Xerox College Publishing, 1972, p. 125 => 1 O18 := limit(x * int(exp(-t^2), t = 0..x)/(1 - exp(-x^2)), x = 0); bytes used=5001676, alloc=1900196, time=6.03 1 time = 0.52, bytes = 325418 # => [-1, 1] O19 := [limit(x/abs(x), x = 0, left), limit(x/abs(x), x = 0, right)]; [-1, 1] time = 0.03, bytes = 21142 # => pi/2 [Richard Q. Chen] O20 := limit(arctan(-log(x)), x = 0, right); 1/2 Pi time = 0.04, bytes = 36158 # ---------- Quit ---------- O21 := quit bytes used=5235136, alloc=1900196, time=6.28 real 7.49 user 6.38 sys 0.99