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/* */
/* Limit operator rule base */
/* */
/* Exponentiation rules */
/* Special limit #1: 0 ^ 0; #2: 1 ^ Infinity; #3: Infinity ^ 0 */
200 # Lim(_var, _tar, _dir, _x ^ _y)_
( [
Local(lx,ly); lx := Lim(var, tar, dir, x); ly := Lim(var, tar, dir, y);
((IsZero(lx) And IsZero(ly)) Or ((lx = 1) And IsInfinity(ly)) Or (IsInfinity(lx) And IsZero(ly)));
] )
<-- Exp(Lim(var, tar, dir, y * Ln(x)));
/* Default rule */
210 # Lim(_var, _tar, _dir, _x ^ _y)
<-- Lim(var, tar, dir, x)^Lim(var, tar, dir, y);
/* Division rules */
/* Special limit #4: 0 / 0; #5: Infinity / Infinity */
300 # Lim(_var, _tar, _dir, _x / _y)_
( [
Local(lx,ly,infx,infy);
lx := Lim(var, tar, dir, x);
ly := Lim(var, tar, dir, y);
infx := (IsInfinity(lx) Or (IsZero(Re(lx)) And IsInfinity(Im(lx))));
infy := (IsInfinity(ly) Or (IsZero(Re(ly)) And IsInfinity(Im(ly))));
((IsZero(lx) And IsZero(ly)) Or
(infx And infy)
);
] )
<-- Lim(var, tar, dir, ApplyPure("D", {var, x})/ApplyPure("D", {var, y}));
/* Special limit #6: null denominator */
/* Probably there are still some problems. */
Dir(Right) <-- 1;
Dir(Left) <-- -1;
/* To get the sign of the denominator on one side: */
Sign(_var, _tar, _dir, _exp, _n)
<-- [
Local(der, coef); der := ApplyPure("D", {var, exp});
coef := Eval(ApplyPure("Subst", {var, tar, der}));
If ( coef = 0,
Sign(var, tar, dir, der, n+1),
(Sign(coef)*Dir(dir)) ^ n
);
];
/* To avoid infinite recursion (with 1/Exp(-x) for instance) */
310 # Lim(_var, _tar, _dir, _x / _y)_
(IsInfinity(tar) And IsZero(Lim(var, tar, dir, y)))
<-- Sign(Lim(var, tar, dir, x))*Sign(Lim(var, tar, dir, ApplyPure("D", {var, y})))*tar;
320 # Lim(_var, _tar, _dir, _x / _y)_IsZero(Lim(var, tar, dir, y))
<-- Sign(Lim(var, tar, dir, x))*Sign(var, tar, dir, y, 1)*Infinity;
/* Default rule */
330 # Lim(_var, _tar, _dir, _x / _y)
<-- Lim(var, tar, dir, x)/Lim(var, tar, dir, y); ];
/* Multiplication rules */
/* To avoid some infinite recursions */
400 # Lim(_var, _tar, _dir, _x * Exp(_y))_
(IsInfinity(Lim(var, tar, dir, x)) And (Lim(var, tar, dir, y) = -Infinity))
<-- Lim(var, tar, dir, x/Exp(-y));
400 # Lim(_var, _tar, _dir, Exp(_x) * _y)_
((Lim(var, tar, dir, x) = -Infinity) And IsInfinity(Lim(var, tar, dir, y)))
<-- Lim(var, tar, dir, y/Exp(-x));
400 # Lim(_var, _tar, _dir, Ln(_x) * _y)_
(IsZero(Lim(var, tar, dir, x)) And IsZero(Lim(var, tar, dir, y)))
<-- Lim(var, tar, dir, y*Ln(x));
/* Special limit #7: 0 * Infinity */
410 # Lim(_var, _tar, _dir, _x * _y)_
((IsZero(Lim(var, tar, dir, x)) And IsInfinity(Lim(var, tar, dir, y)))
Or (IsInfinity(Lim(var, tar, dir, x)) And IsZero(Lim(var, tar, dir, y))))
<-- Lim(var, tar, dir, Simplify(ApplyPure("D", {var, y})/ApplyPure("D",
{var, 1/x})));
/* Default rule */
420 # Lim(_var, _tar, _dir, _x * _y)
<-- Lim(var, tar, dir, x) * Lim(var, tar, dir, y);
/* Substraction rules */
/* Special limit #8: Infinity - Infinity */
500 # Lim(_var, _tar, _dir, _x - _y)_
( [
Local(lx,ly); lx := Lim(var, tar, dir, x); ly := Lim(var, tar, dir, y);
((lx = Infinity) And (ly = Infinity)) Or ((lx = -Infinity) And (ly = -Infinity));
] )
<-- Lim(var, tar, dir, x*(1-y/x));
/* Default rule */
510 # Lim(_var, _tar, _dir, _x - _y)
<-- Lim(var, tar, dir, x)-Lim(var, tar, dir, y);
/* Unary minus */
520 # Lim(_var, _tar, _dir, - _x)
<-- - Lim(var, tar, dir, x);
/* Addition rules */
/* Special limit #9: Infinity + (-Infinity) */
600 # Lim(_var, _tar, _dir, _x + _y)_
( [
Local(lx,ly); lx := Lim(var, tar, dir, x); ly := Lim(var, tar, dir, y);
((lx = Infinity) And (ly = -Infinity)) Or ((lx = -Infinity) And (ly = Infinity));
] )
<-- Lim(var, tar, dir, x*(1+y/x));
/* Default rule */
610 # Lim(_var, _tar, _dir, _x + _y)
<-- Lim(var, tar, dir, x)+Lim(var, tar, dir, y);
/* Global default rule : evaluate expression */
700 # Lim(_var, _tar, _dir, exp_IsFunction)
<-- Eval(MapArgs(exp,"LimitArgs"));
LimitArgs(_arg) <-- Lim(var,tar,dir,arg);
UnFence("LimitArgs",1); /* Allow LimitArgs to have access to the local variables of the caller. */
701 # Lim(_var, _tar, _dir, _exp)
<-- Eval(ApplyPure("Subst", {var, tar, exp}));
/* Limit without direction */
10 # Lim(_var, tar_IsInfinity, _exp) <-- Lim(var, tar, None, exp);
20 # Lim(_var, _tar, _exp)
<-- [
Local(l); l := Lim(var, tar, Left, exp);
If ( l = Lim(var, tar, Right, exp),
l,
Undefined
);
];
/* User-callable function */
(Limit(_var,_lim)(_fie)) <-- Lim(var,lim,fie);
(Limit(_var,_lim,_direction)(_fie)) <-- Lim(var,lim,direction,fie);
UnFence("Limit",3);
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