Artifact 693b914fc047b2f1ff2eeaa2df7020de8e7bbcc7806126530ee32663827621b4:
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r36/xlog/LIMITS.LOG
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2011-09-02 18:13:33
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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)