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Sat Jun 29 13:47:50 PDT 1991 REDUCE 3.4, 15-Jul-91 ... 1: 1: 2: 2: 3: 3: % Tests of limits package. limit(sin(x)/x,x,0); 1 limit(sin(x)^2/x,x,0); 0 limit(sin(x)/x,x,1); SIN(1) limit(1/x,x,0); INFINITY limit(-1/x,x,0); - INFINITY limit((sin(x)-x)/x^3,x,0); - 1 ------ 6 limit(x*sin(1/x),x,infinity); 1 limit(sin x/x^2,x,0); INFINITY limit(x^2*sin(1/x),x,infinity); INFINITY % Simple examples from Schaum's Theory & Problems of Advanced Calculus limit(x^2-6x+4,x,2); -4 limit((x+3)*(2x-1)/(x^2+3x-2),x,-1); 3 --- 2 limit((sqrt(4+h)-2)/h,h,0); 1 --- 4 limit((sqrt(x)-2)/(4-x),x,4); - 1 ------ 4 limit((x^2-4)/(x-2),x,2); 4 limit(1/(2x-5),x,-1); - 1 ------ 7 limit(sqrt(x)/(x+1),x,1); 1 --- 2 limit((2x+5)/(3x-2),x,infinity); 2 --- 3 limit((1/(x+3)-2/(3x+5))/(x-1),x,1); 1 ---- 32 limit(sin(3x)/x,x,0); 3 limit((1-cos(x))/x^2,x,0); 1 --- 2 limit((6x-sin(2x))/(2x+3*sin(4x)),x,0); 2 --- 7 limit((1-2*cos(x)+cos(2x))/x^2,x,0); -1 limit((3*sin(pi*x) - sin(3*pi*x))/x^3,x,0); 3 4*PI limit((cos(a*x)-cos(b*x))/x^2,x,0); 2 2 - A + B ------------ 2 limit((e^x-1)/x,x,0); 1 limit((a^x-b^x)/x,x,0); LOG(A) - LOG(B) % Examples taken from Hyslop's Real Variable limit(sinh(2x)^2/log(1+x^2),x,0); 4 limit(x^2*(e^(1/x)-1)*(log(x+2)-log(x)),x,infinity); 2 limit(x^alpha*log(x+1)^2/log(x),x,infinity); FAILED %% 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 limit((x*sinh(x)-2+2*cosh(x))/(x^4+2*x^2),x,0); 1 limit((2*sinh(x)-tanh(x))/(e^x-1),x,0); 1 limit(x*tanh(x)/(sqrt(1-x^2)-1),x,0); -2 limit((2*log(1+x)+x^2-2*x)/x^3,x,0); 2 --- 3 limit((e^(5*x)-2*x)^(1/x),x,0); 3 E limit(log(log(x))/log(x)^2,x,infinity); 0 % These are adapted from Lession 4 from Stoutmyer limit((e^x-1)/x, x, 0); 1 limit(((1-x)/log(x))**2, x, 1); 1 limit(x/(e**x-1), x, 0); 1 %% One sided limits limit!+(sin(x)/sqrt(x),x,0); 0 limit!-(sin(x)/sqrt(x),x,0); 0 limit(x/log x,x,0); 0 limit(log(1 + x)/log x,x,infinity); 1 limit(log x/sqrt x,x,infinity); 0 limit!+(sqrt x/sin x,x,0); INFINITY limit(log x,x,0); - INFINITY limit(x*log x,x,0); 0 limit(log x/log(2x),x,0); 1 limit(log x*log(1+x)*(1+x),x,0); 0 limit(log x/x,x,infinity); 0 limit(log x/sqrt x,x,infinity); 0 limit(log x,x,infinity); INFINITY limit(log(x+1)/sin x,x,0); 1 limit(log(1+1/x)*sin x,x,0); 0 limit(-log(1+x)*(x+2)/sin x,x,0); -2 limit(-log x*(3+x)/log(2x),x,0); -3 limit(log(x+1)^2/sqrt x,x,infinity); 0 limit(log(x + 1) - log x,x,infinity); 0 limit(-(log x)^2/log log x,x,infinity); - INFINITY limit(log(x-1)/sin x,x,0); INFINITY %% -> INFINITY, but what should it be? limit!-(sqrt x/sin x,x,0); INFINITY % infinity limit(log x-log(2x),x,0); - LOG(2) % or any other limit! limit(sqrt x-sqrt(x+1),x,infinity); 0 limit(sin sin x/x,x,0); 1 limit!-(sin x/cos x,x,pi/2); INFINITY % this works! limit!+(sin x/cos x,x,pi/2); - INFINITY % so does this! % but limit!+(tan x,x,pi/2) fails unless tan is defined using let. limit(sin x/cosh x,x,infinity); 0 limit(sin x/x,x,infinity); 0 limit(x*sin(1/x),x,0); 0 limit(exp x/((exp x + exp(-x))/2),x,infinity); 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 limit(log x*sin(x^2)/(x*sinh x),x,0); - INFINITY limit(sin(x^2)/(x*sinh x*log x),x,0); 0 limit(log x/log(x^2),x,0); 1 --- 2 limit(log(x^2)-log(x^2+8x),x,0); - INFINITY limit(log(x^2)-log(x^2+8x),x,infinity); 0 limit(sqrt(x+5)-sqrt x,x,infinity); 0 limit(2^(log x),x,0); 0 limit((sin tan x-tan sin x)/(asin atan x-atan asin x),x,0); 1 % This one has the value infinity, but fails with de L'Hospital's rule: limit((e+1)^(x^2)/e^x,x,infinity); FAILED showtime; Time: 42755 ms end; 4: 4: Quitting Sat Jun 29 13:48:36 PDT 1991