Codemist Standard Lisp 3.54 for DEC Alpha: May 23 1994
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% 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);
infinity
% infinity
limit!-(sqrt x/sin x,x,0);
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);
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 1/3
3*x *(3*sqrt(4*x + 27)*sqrt(3) - 2*x - 27)
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);
1/3
- 4
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);
1/5
- 216
-----------
1/5
7
% -6^(3/5)/7^(1/5)
end;
(TIME: limits 120478 125594)
End of Lisp run after 120.51+5.74 seconds