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