@@ -1,832 +1,832 @@ -REDUCE 3.6, 15-Jul-95, patched to 6 Mar 96 ... - - -% Tests of limits package. - -limit(sin(x)/x,x,0); - - -1 - % 1 -limit(sin(x)^2/x,x,0); - - -0 - % 0 -limit(sin(x)/x,x,1); - - -sin(1) - % sin(1) -limit(1/x,x,0); - - -infinity - % infinity -limit(-1/x,x,0); - - - - infinity - % - infinity -limit((sin(x)-x)/x^3,x,0); - - - - 1 ------- - 6 - % -1/6 -limit(x*sin(1/x),x,infinity); - - -1 - % 1 -limit(sin x/x^2,x,0); - - -infinity - % infinity -limit(x^2*sin(1/x),x,infinity); - - -infinity - % infinity - -% Simple examples from Schaum's Theory & Problems of Advanced Calculus - -limit(x^2-6x+4,x,2); - - --4 - % -4 -limit((x+3)*(2x-1)/(x^2+3x-2),x,-1); - - - 3 ---- - 2 - % 3/2 -limit((sqrt(4+h)-2)/h,h,0); - - - 1 ---- - 4 - % 1/4 -limit((sqrt(x)-2)/(4-x),x,4); - - - - 1 ------- - 4 - % -1/4 -limit((x^2-4)/(x-2),x,2); - - -4 - % 4 -limit(1/(2x-5),x,-1); - - - - 1 ------- - 7 - % -1/7 -limit(sqrt(x)/(x+1),x,1); - - - 1 ---- - 2 - % 1/2 -limit((2x+5)/(3x-2),x,infinity); - - - 2 ---- - 3 - % 2/3 -limit((1/(x+3)-2/(3x+5))/(x-1),x,1); - - - 1 ----- - 32 - % 1/32 -limit(sin(3x)/x,x,0); - - -3 - % 3 -limit((1-cos(x))/x^2,x,0); - - - 1 ---- - 2 - % 1/2 -limit((6x-sin(2x))/(2x+3*sin(4x)),x,0); - - - 2 ---- - 7 - % 2/7 -limit((1-2*cos(x)+cos(2x))/x^2,x,0); - - --1 - % -1 -limit((3*sin(pi*x) - sin(3*pi*x))/x^3,x,0); - - - 3 -4*pi - % 4*pi^3 -limit((cos(a*x)-cos(b*x))/x^2,x,0); - - - 2 2 - - a + b ------------- - 2 - % (-a^2 + b^2)/2 -limit((e^x-1)/x,x,0); - - -1 - % 1 -limit((a^x-b^x)/x,x,0); - - -log(a) - log(b) - % log(a) - log(b) - -% Examples taken from Hyslop's Real Variable - -limit(sinh(2x)^2/log(1+x^2),x,0); - - -4 - % 4 -limit(x^2*(e^(1/x)-1)*(log(x+2)-log(x)),x,infinity); - - -2 - % 2 -limit(x^alpha*log(x+1)^2/log(x),x,infinity); - - - 2 - alpha log(x + 1) -limit(x *-------------,x,infinity) - log(x) - - %% if repart alpha < 0 then 0 else infinity. - %% fails because answer depends in essential way on parameter. - -limit((2*cosh(x)-2-x^2)/log(1+x^2)^2,x,0); - - - 1 ----- - 12 - % 1/12 -limit((x*sinh(x)-2+2*cosh(x))/(x^4+2*x^2),x,0); - - -1 - % 1 -limit((2*sinh(x)-tanh(x))/(e^x-1),x,0); - - -1 - % 1 -limit(x*tanh(x)/(sqrt(1-x^2)-1),x,0); - - --2 - % -2 -limit((2*log(1+x)+x^2-2*x)/x^3,x,0); - - - 2 ---- - 3 - % 2/3 -limit((e^(5*x)-2*x)^(1/x),x,0); - - - 3 -e - % e^3 -limit(log(log(x))/log(x)^2,x,infinity); - - -0 - % 0 - -% These are adapted from Lession 4 from Stoutmyer - -limit((e^x-1)/x, x, 0); - - -1 - % 1 -limit(((1-x)/log(x))**2, x, 1); - - -1 - % 1 -limit(x/(e**x-1), x, 0); - - -1 - % 1 - -%% One sided limits -limit!+(sin(x)/sqrt(x),x,0); - - -0 - % 0 -limit!-(sin(x)/sqrt(x),x,0); - - -0 - % 0 - - -limit(x/log x,x,0); - - -0 - % 0 -limit(log(1 + x)/log x,x,infinity); - - -1 - % 1 -limit(log x/sqrt x,x,infinity); - - -0 - % 0 -limit!+(sqrt x/sin x,x,0); - - -infinity - % infinity -limit(log x,x,0); - - - - infinity - % - infinity -limit(x*log x,x,0); - - -0 - % 0 -limit(log x/log(2x),x,0); - - -1 - % 1 -limit(log x*log(1+x)*(1+x),x,0); - - -0 - % 0 -limit(log x/x,x,infinity); - - -0 - % 0 -limit(log x/sqrt x,x,infinity); - - -0 - % 0 -limit(log x,x,infinity); - - -infinity - % infinity -limit(log(x+1)/sin x,x,0); - - -1 - % 1 -limit(log(1+1/x)*sin x,x,0); - - -0 - % 0 -limit(-log(1+x)*(x+2)/sin x,x,0); - - --2 - % -2 -limit(-log x*(3+x)/log(2x),x,0); - - --3 - % -3 -limit(log(x+1)^2/sqrt x,x,infinity); - - -0 - % 0 -limit(log(x + 1) - log x,x,infinity); - - -0 - % 0 -limit(-(log x)^2/log log x,x,infinity); - - - - infinity - % - infinity -limit(log(x-1)/sin x,x,0); - - -sign(log(-1))*infinity - % infinity - -limit!-(sqrt x/sin x,x,0); - - - - sign(i)*infinity - % infinity -limit(log x-log(2x),x,0); - - - - log(2) - % - log(2) -limit(sqrt x-sqrt(x+1),x,infinity); - - -0 - % 0 - -limit(sin sin x/x,x,0); - - -1 - % 1 -limit!-(sin x/cos x,x,pi/2); - - -infinity - % infinity % this works! -limit!+(sin x/cos x,x,pi/2); - - - - infinity - % - infinity % so does this! -limit(sin x/cosh x,x,infinity); - - -0 - % 0 -limit(sin x/x,x,infinity); - - -0 - % 0 -limit(x*sin(1/x),x,0); - - -0 - % 0 -limit(exp x/((exp x + exp(-x))/2),x,infinity); - - -2 - % 2 -% limit(exp x/cosh x,x,infinity); % fails in this form, but if cosh is - %defined using let, then it works. -limit((sin(x^2)/(x*sinh x)),x,0); - - -1 - % 1 -limit(log x*sin(x^2)/(x*sinh x),x,0); - - - - infinity - % - infinity -limit(sin(x^2)/(x*sinh x*log x),x,0); - - -0 - % 0 -limit(log x/log(x^2),x,0); - - - 1 ---- - 2 - % 1/2 -limit(log(x^2)-log(x^2+8x),x,0); - - - - infinity - % - infinity -limit(log(x^2)-log(x^2+8x),x,infinity); - - -0 - % 0 -limit(sqrt(x+5)-sqrt x,x,infinity); - - -0 - % 0 -limit(2^(log x),x,0); - - -0 - % 0 - -% Additional examples -limit((sin tan x-tan sin x)/(asin atan x-atan asin x),x,0); - - -1 - % 1 - -% This one has the value infinity, but fails with de L'Hospital's rule: -limit((e+1)^(x^2)/e^x,x,infinity); - - - 2 - x - (e + 1) -limit(-----------,x,infinity) - x - e - % infinity % fails - -comment -The following examples were not in the previous set$ - - -% Simon test examples: -limit(log(x-a)/((a-b)*(a-c)) + log(2(x-b))/((b-c)*(b-a)) - + log(x-c)/((c-a)*(c-b)),x,infinity); - - - 1 - log(---) - 2 ----------------------- - 2 - a*b - a*c - b + b*c - % log(1/2)/((a-b)*(b-c)) - -limit(1/(e^x-e^(x-1/x^2)),x,infinity); - - - 1 -limit(----------------,x,infinity) - 2 - x x - 1/x - e - e - % infinity % fails - -% new capabilities: branch points at the origin, needed for definite -% integration. - -limit(x+sqrt x,x,0); - - -0 - % 0 -limit!+(sqrt x/(x+1),x,0); - - -0 - % 0 -limit!+(x^(1/3)/(x+1),x,0); - - -0 - % 0 -limit(log(x)^2/x^(1/3),x,0); - - -infinity - % infinity -limit(log x/x^(1/3),x,0); - - - - infinity - % - infinity - -h := (X^(1/3) + 3*X**(1/4))/(7*(SQRT(X + 9) - 3)**(1/4)); - - - 1/4 1/3 - 3*x + x -h := ------------------------ - 1/4 - 7*(sqrt(x + 9) - 3) - -limit(h,x,0); - - - 1/4 - 3*6 --------- - 7 - % 3/7*6^(1/4) - -% Examples from Paul S. Wang's thesis: - -limit(x^log(1/x),x,infinity); - - -0 - % 0 -limit(cos x - 1/(e^x^2 - 1),x,0); - - - - infinity - % - infinity -limit((1+a*x)^(1/x),x,infinity); - - -1 - % 1 -limit(x^2*sqrt(4*x^4+5)-2*x^4,x,infinity); - - - 5 ---- - 4 - % 5/4 -limit!+(1/x-1/sin x,x,0); - - -0 - % 0 -limit(e^(x*sqrt(x^2+1))-e^(x^2),x,infinity); - - - 2 2 - x*sqrt(x + 1) x -limit(e - e ,x,infinity) - % 0 fails -limit((e^x+x*log x)/(log(x^4+x+1)+e^sqrt(x^3+1)),x,infinity); - - - x - e + x*log(x) -limit(---------------------------------,x,infinity) - 3 - 4 sqrt(x + 1) - log(x + x + 1) + e - %0 % fails -limit!-(1/(x^3-6*x+11*x-6),x,2); - - - 1 ----- - 12 - % 1/12 -limit((x*sqrt(x+5))/(sqrt(4*x^3+1)+x),x,infinity); - - - 1 ---- - 2 - % 1/2 -limit!-(tan x/log cos x,x,pi/2); - - - - infinity - % - infinity - -z0 := z*(z-2*pi*i)*(z-pi*i/2)/(sinh z - i); - - - 2 2 - z*( - 5*i*pi*z - 2*pi + 2*z ) -z0 := -------------------------------- - 2*(sinh(z) - i) - -limit(df(z0,z),z,pi*i/2); - - -sign(i)*infinity - % infinity -z1 := z0*(z-pi*i/2); - - - 3 2 2 3 - z*(2*i*pi - 12*i*pi*z - 9*pi *z + 4*z ) -z1 := ------------------------------------------- - 4*(sinh(z) - i) - -limit(df(z1,z),z,pi*i/2); - - - - 2*pi - % -2*pi - -% and the analogous problem: -z2 := z*(z-2*pi)*(z-pi/2)/(sin z - 1); - - - 2 2 - z*(2*pi - 5*pi*z + 2*z ) -z2 := --------------------------- - 2*(sin(z) - 1) - -limit(df(z2,z),z,pi/2); - - - - infinity - % infinity -z3 := z2*(z-pi/2); - - - 3 2 2 3 - z*( - 2*pi + 9*pi *z - 12*pi*z + 4*z ) -z3 := ------------------------------------------ - 4*(sin(z) - 1) - -limit(df(z3,z),z,pi/2); - - -2*pi - % 2*pi - -% A test by Wolfram Koepf. -f:=x^2/(3*(-27*x^2 - 2*x^3 + 3^(3/2)*(27*x^4 + 4*x^5)^(1/2))^(1/3)); - - - 2 - x -f := -------------------------------------------------------- - 2 3 2 1/3 - 3*(3*sqrt(4*x + 27)*sqrt(3)*abs(x) - 2*x - 27*x ) - -L0:=limit(f,x,0); - - -l0 := 0 - % L0 := 0 -f1:=((f-L0)/x^(1/3))$ - - -L1:=limit(f1,x,0); - - -l1 := 0 - % L1 := 0 -f2:=((f1-L1)/x^(1/3))$ - - -L2:=limit(f2,x,0); - - - - 1 -l2 := ------ - 1/3 - 2 - % L2 := -1/2^(1/3) -f3:=((f2-L2)/x^(1/3))$ - - -L3:=limit(f3,x,0); - - -l3 := 0 - % L3 := 0 -f4:=((f3-L3)/x^(1/3))$ - - -L4:=limit(f4,x,0); - - -l4 := 0 - % L4 := 0 -f5:=((f4-L4)/x^(1/3))$ - - -L5:=limit(f5,x,0); - - - 2/3 - - 2 -l5 := --------- - 81 - % L5 = -2^(2/3)/81 -f6:=((f5-L5)/x^(1/3))$ - - -L6:=limit(f6,x,0); - - -l6 := 0 - % L6 := 0 -f7:=((f6-L6)/x^(1/3))$ - - -L7:=limit(f7,x,0); - - -l7 := 0 - % L7 := 0 -f8:=((f7-L7)/x^(1/3))$ - - -L8:=limit(f8,x,0); - - - 7 -l8 := ----------- - 1/3 - 6561*2 - % L8 := 7/(6561*2^(1/3)) - -limit(log(1+x)^2/x^(1/3),x,infinity); - - -0 - % 0 -limit(e^(log(1+x)^2/x^(1/3)),x,infinity); - - -1 - % 1 - -ss := (sqrt(x^(2/5) +1) - x^(1/3)-1)/x^(1/3); - - - 2/5 1/3 - sqrt(x + 1) - x - 1 -ss := --------------------------- - 1/3 - x - -limit(ss,x,0); - - --1 - % -1 -limit(exp(ss),x,0); - - - 1 ---- - e - % 1/e -limit(log x,x,-1); - - -log(-1) - % log(-1) -limit(log(ss),x,0); - - -log(-1) - % log(-1) - -ss := ((x^(1/2) - 1)^(1/3) + (x^(1/5) + 1)^2)/x^(1/5); - - - 1/3 2/5 1/5 - (sqrt(x) - 1) + x + 2*x + 1 -ss := -------------------------------------- - 1/5 - x - -limit(ss,x,0); - - -2 - % 2 - -h := (X^(1/5) + 3*X**(1/4))^2/(7*(SQRT(X + 9) - 3 - x/6))**(1/5); - - - 1/5 2/5 9/20 - 6 *(x + 6*x + 9*sqrt(x)) -h := ----------------------------------- - 1/5 1/5 - (6*sqrt(x + 9) - x - 18) *7 - -limit(h,x,0); - - - 3/5 - - 6 ---------- - 1/5 - 7 - % -6^(3/5)/7^(1/5) - -end; -(TIME: limits 28710 30110) +REDUCE 3.6, 15-Jul-95, patched to 6 Mar 96 ... + + +% Tests of limits package. + +limit(sin(x)/x,x,0); + + +1 + % 1 +limit(sin(x)^2/x,x,0); + + +0 + % 0 +limit(sin(x)/x,x,1); + + +sin(1) + % sin(1) +limit(1/x,x,0); + + +infinity + % infinity +limit(-1/x,x,0); + + + - infinity + % - infinity +limit((sin(x)-x)/x^3,x,0); + + + - 1 +------ + 6 + % -1/6 +limit(x*sin(1/x),x,infinity); + + +1 + % 1 +limit(sin x/x^2,x,0); + + +infinity + % infinity +limit(x^2*sin(1/x),x,infinity); + + +infinity + % infinity + +% Simple examples from Schaum's Theory & Problems of Advanced Calculus + +limit(x^2-6x+4,x,2); + + +-4 + % -4 +limit((x+3)*(2x-1)/(x^2+3x-2),x,-1); + + + 3 +--- + 2 + % 3/2 +limit((sqrt(4+h)-2)/h,h,0); + + + 1 +--- + 4 + % 1/4 +limit((sqrt(x)-2)/(4-x),x,4); + + + - 1 +------ + 4 + % -1/4 +limit((x^2-4)/(x-2),x,2); + + +4 + % 4 +limit(1/(2x-5),x,-1); + + + - 1 +------ + 7 + % -1/7 +limit(sqrt(x)/(x+1),x,1); + + + 1 +--- + 2 + % 1/2 +limit((2x+5)/(3x-2),x,infinity); + + + 2 +--- + 3 + % 2/3 +limit((1/(x+3)-2/(3x+5))/(x-1),x,1); + + + 1 +---- + 32 + % 1/32 +limit(sin(3x)/x,x,0); + + +3 + % 3 +limit((1-cos(x))/x^2,x,0); + + + 1 +--- + 2 + % 1/2 +limit((6x-sin(2x))/(2x+3*sin(4x)),x,0); + + + 2 +--- + 7 + % 2/7 +limit((1-2*cos(x)+cos(2x))/x^2,x,0); + + +-1 + % -1 +limit((3*sin(pi*x) - sin(3*pi*x))/x^3,x,0); + + + 3 +4*pi + % 4*pi^3 +limit((cos(a*x)-cos(b*x))/x^2,x,0); + + + 2 2 + - a + b +------------ + 2 + % (-a^2 + b^2)/2 +limit((e^x-1)/x,x,0); + + +1 + % 1 +limit((a^x-b^x)/x,x,0); + + +log(a) - log(b) + % log(a) - log(b) + +% Examples taken from Hyslop's Real Variable + +limit(sinh(2x)^2/log(1+x^2),x,0); + + +4 + % 4 +limit(x^2*(e^(1/x)-1)*(log(x+2)-log(x)),x,infinity); + + +2 + % 2 +limit(x^alpha*log(x+1)^2/log(x),x,infinity); + + + 2 + alpha log(x + 1) +limit(x *-------------,x,infinity) + log(x) + + %% if repart alpha < 0 then 0 else infinity. + %% fails because answer depends in essential way on parameter. + +limit((2*cosh(x)-2-x^2)/log(1+x^2)^2,x,0); + + + 1 +---- + 12 + % 1/12 +limit((x*sinh(x)-2+2*cosh(x))/(x^4+2*x^2),x,0); + + +1 + % 1 +limit((2*sinh(x)-tanh(x))/(e^x-1),x,0); + + +1 + % 1 +limit(x*tanh(x)/(sqrt(1-x^2)-1),x,0); + + +-2 + % -2 +limit((2*log(1+x)+x^2-2*x)/x^3,x,0); + + + 2 +--- + 3 + % 2/3 +limit((e^(5*x)-2*x)^(1/x),x,0); + + + 3 +e + % e^3 +limit(log(log(x))/log(x)^2,x,infinity); + + +0 + % 0 + +% These are adapted from Lession 4 from Stoutmyer + +limit((e^x-1)/x, x, 0); + + +1 + % 1 +limit(((1-x)/log(x))**2, x, 1); + + +1 + % 1 +limit(x/(e**x-1), x, 0); + + +1 + % 1 + +%% One sided limits +limit!+(sin(x)/sqrt(x),x,0); + + +0 + % 0 +limit!-(sin(x)/sqrt(x),x,0); + + +0 + % 0 + + +limit(x/log x,x,0); + + +0 + % 0 +limit(log(1 + x)/log x,x,infinity); + + +1 + % 1 +limit(log x/sqrt x,x,infinity); + + +0 + % 0 +limit!+(sqrt x/sin x,x,0); + + +infinity + % infinity +limit(log x,x,0); + + + - infinity + % - infinity +limit(x*log x,x,0); + + +0 + % 0 +limit(log x/log(2x),x,0); + + +1 + % 1 +limit(log x*log(1+x)*(1+x),x,0); + + +0 + % 0 +limit(log x/x,x,infinity); + + +0 + % 0 +limit(log x/sqrt x,x,infinity); + + +0 + % 0 +limit(log x,x,infinity); + + +infinity + % infinity +limit(log(x+1)/sin x,x,0); + + +1 + % 1 +limit(log(1+1/x)*sin x,x,0); + + +0 + % 0 +limit(-log(1+x)*(x+2)/sin x,x,0); + + +-2 + % -2 +limit(-log x*(3+x)/log(2x),x,0); + + +-3 + % -3 +limit(log(x+1)^2/sqrt x,x,infinity); + + +0 + % 0 +limit(log(x + 1) - log x,x,infinity); + + +0 + % 0 +limit(-(log x)^2/log log x,x,infinity); + + + - infinity + % - infinity +limit(log(x-1)/sin x,x,0); + + +sign(log(-1))*infinity + % infinity + +limit!-(sqrt x/sin x,x,0); + + + - sign(i)*infinity + % infinity +limit(log x-log(2x),x,0); + + + - log(2) + % - log(2) +limit(sqrt x-sqrt(x+1),x,infinity); + + +0 + % 0 + +limit(sin sin x/x,x,0); + + +1 + % 1 +limit!-(sin x/cos x,x,pi/2); + + +infinity + % infinity % this works! +limit!+(sin x/cos x,x,pi/2); + + + - infinity + % - infinity % so does this! +limit(sin x/cosh x,x,infinity); + + +0 + % 0 +limit(sin x/x,x,infinity); + + +0 + % 0 +limit(x*sin(1/x),x,0); + + +0 + % 0 +limit(exp x/((exp x + exp(-x))/2),x,infinity); + + +2 + % 2 +% limit(exp x/cosh x,x,infinity); % fails in this form, but if cosh is + %defined using let, then it works. +limit((sin(x^2)/(x*sinh x)),x,0); + + +1 + % 1 +limit(log x*sin(x^2)/(x*sinh x),x,0); + + + - infinity + % - infinity +limit(sin(x^2)/(x*sinh x*log x),x,0); + + +0 + % 0 +limit(log x/log(x^2),x,0); + + + 1 +--- + 2 + % 1/2 +limit(log(x^2)-log(x^2+8x),x,0); + + + - infinity + % - infinity +limit(log(x^2)-log(x^2+8x),x,infinity); + + +0 + % 0 +limit(sqrt(x+5)-sqrt x,x,infinity); + + +0 + % 0 +limit(2^(log x),x,0); + + +0 + % 0 + +% Additional examples +limit((sin tan x-tan sin x)/(asin atan x-atan asin x),x,0); + + +1 + % 1 + +% This one has the value infinity, but fails with de L'Hospital's rule: +limit((e+1)^(x^2)/e^x,x,infinity); + + + 2 + x + (e + 1) +limit(-----------,x,infinity) + x + e + % infinity % fails + +comment +The following examples were not in the previous set$ + + +% Simon test examples: +limit(log(x-a)/((a-b)*(a-c)) + log(2(x-b))/((b-c)*(b-a)) + + log(x-c)/((c-a)*(c-b)),x,infinity); + + + 1 + log(---) + 2 +---------------------- + 2 + a*b - a*c - b + b*c + % log(1/2)/((a-b)*(b-c)) + +limit(1/(e^x-e^(x-1/x^2)),x,infinity); + + + 1 +limit(----------------,x,infinity) + 2 + x x - 1/x + e - e + % infinity % fails + +% new capabilities: branch points at the origin, needed for definite +% integration. + +limit(x+sqrt x,x,0); + + +0 + % 0 +limit!+(sqrt x/(x+1),x,0); + + +0 + % 0 +limit!+(x^(1/3)/(x+1),x,0); + + +0 + % 0 +limit(log(x)^2/x^(1/3),x,0); + + +infinity + % infinity +limit(log x/x^(1/3),x,0); + + + - infinity + % - infinity + +h := (X^(1/3) + 3*X**(1/4))/(7*(SQRT(X + 9) - 3)**(1/4)); + + + 1/4 1/3 + 3*x + x +h := ------------------------ + 1/4 + 7*(sqrt(x + 9) - 3) + +limit(h,x,0); + + + 1/4 + 3*6 +-------- + 7 + % 3/7*6^(1/4) + +% Examples from Paul S. Wang's thesis: + +limit(x^log(1/x),x,infinity); + + +0 + % 0 +limit(cos x - 1/(e^x^2 - 1),x,0); + + + - infinity + % - infinity +limit((1+a*x)^(1/x),x,infinity); + + +1 + % 1 +limit(x^2*sqrt(4*x^4+5)-2*x^4,x,infinity); + + + 5 +--- + 4 + % 5/4 +limit!+(1/x-1/sin x,x,0); + + +0 + % 0 +limit(e^(x*sqrt(x^2+1))-e^(x^2),x,infinity); + + + 2 2 + x*sqrt(x + 1) x +limit(e - e ,x,infinity) + % 0 fails +limit((e^x+x*log x)/(log(x^4+x+1)+e^sqrt(x^3+1)),x,infinity); + + + x + e + x*log(x) +limit(---------------------------------,x,infinity) + 3 + 4 sqrt(x + 1) + log(x + x + 1) + e + %0 % fails +limit!-(1/(x^3-6*x+11*x-6),x,2); + + + 1 +---- + 12 + % 1/12 +limit((x*sqrt(x+5))/(sqrt(4*x^3+1)+x),x,infinity); + + + 1 +--- + 2 + % 1/2 +limit!-(tan x/log cos x,x,pi/2); + + + - infinity + % - infinity + +z0 := z*(z-2*pi*i)*(z-pi*i/2)/(sinh z - i); + + + 2 2 + z*( - 5*i*pi*z - 2*pi + 2*z ) +z0 := -------------------------------- + 2*(sinh(z) - i) + +limit(df(z0,z),z,pi*i/2); + + +sign(i)*infinity + % infinity +z1 := z0*(z-pi*i/2); + + + 3 2 2 3 + z*(2*i*pi - 12*i*pi*z - 9*pi *z + 4*z ) +z1 := ------------------------------------------- + 4*(sinh(z) - i) + +limit(df(z1,z),z,pi*i/2); + + + - 2*pi + % -2*pi + +% and the analogous problem: +z2 := z*(z-2*pi)*(z-pi/2)/(sin z - 1); + + + 2 2 + z*(2*pi - 5*pi*z + 2*z ) +z2 := --------------------------- + 2*(sin(z) - 1) + +limit(df(z2,z),z,pi/2); + + + - infinity + % infinity +z3 := z2*(z-pi/2); + + + 3 2 2 3 + z*( - 2*pi + 9*pi *z - 12*pi*z + 4*z ) +z3 := ------------------------------------------ + 4*(sin(z) - 1) + +limit(df(z3,z),z,pi/2); + + +2*pi + % 2*pi + +% A test by Wolfram Koepf. +f:=x^2/(3*(-27*x^2 - 2*x^3 + 3^(3/2)*(27*x^4 + 4*x^5)^(1/2))^(1/3)); + + + 2 + x +f := -------------------------------------------------------- + 2 3 2 1/3 + 3*(3*sqrt(4*x + 27)*sqrt(3)*abs(x) - 2*x - 27*x ) + +L0:=limit(f,x,0); + + +l0 := 0 + % L0 := 0 +f1:=((f-L0)/x^(1/3))$ + + +L1:=limit(f1,x,0); + + +l1 := 0 + % L1 := 0 +f2:=((f1-L1)/x^(1/3))$ + + +L2:=limit(f2,x,0); + + + - 1 +l2 := ------ + 1/3 + 2 + % L2 := -1/2^(1/3) +f3:=((f2-L2)/x^(1/3))$ + + +L3:=limit(f3,x,0); + + +l3 := 0 + % L3 := 0 +f4:=((f3-L3)/x^(1/3))$ + + +L4:=limit(f4,x,0); + + +l4 := 0 + % L4 := 0 +f5:=((f4-L4)/x^(1/3))$ + + +L5:=limit(f5,x,0); + + + 2/3 + - 2 +l5 := --------- + 81 + % L5 = -2^(2/3)/81 +f6:=((f5-L5)/x^(1/3))$ + + +L6:=limit(f6,x,0); + + +l6 := 0 + % L6 := 0 +f7:=((f6-L6)/x^(1/3))$ + + +L7:=limit(f7,x,0); + + +l7 := 0 + % L7 := 0 +f8:=((f7-L7)/x^(1/3))$ + + +L8:=limit(f8,x,0); + + + 7 +l8 := ----------- + 1/3 + 6561*2 + % L8 := 7/(6561*2^(1/3)) + +limit(log(1+x)^2/x^(1/3),x,infinity); + + +0 + % 0 +limit(e^(log(1+x)^2/x^(1/3)),x,infinity); + + +1 + % 1 + +ss := (sqrt(x^(2/5) +1) - x^(1/3)-1)/x^(1/3); + + + 2/5 1/3 + sqrt(x + 1) - x - 1 +ss := --------------------------- + 1/3 + x + +limit(ss,x,0); + + +-1 + % -1 +limit(exp(ss),x,0); + + + 1 +--- + e + % 1/e +limit(log x,x,-1); + + +log(-1) + % log(-1) +limit(log(ss),x,0); + + +log(-1) + % log(-1) + +ss := ((x^(1/2) - 1)^(1/3) + (x^(1/5) + 1)^2)/x^(1/5); + + + 1/3 2/5 1/5 + (sqrt(x) - 1) + x + 2*x + 1 +ss := -------------------------------------- + 1/5 + x + +limit(ss,x,0); + + +2 + % 2 + +h := (X^(1/5) + 3*X**(1/4))^2/(7*(SQRT(X + 9) - 3 - x/6))**(1/5); + + + 1/5 2/5 9/20 + 6 *(x + 6*x + 9*sqrt(x)) +h := ----------------------------------- + 1/5 1/5 + (6*sqrt(x + 9) - x - 18) *7 + +limit(h,x,0); + + + 3/5 + - 6 +--------- + 1/5 + 7 + % -6^(3/5)/7^(1/5) + +end; +(TIME: limits 28710 30110)