Comment #17 on issue 835 by [email protected]: limit((5**x+3**x)**(1/x), x,
oo) TODO
http://code.google.com/p/sympy/issues/detail?id=835
Perhaps the problem is elsewhere:
These are ok for x's exponent <1, =1, >1
h[4] >>> limit(((1+sqrt(x))**(1/x)),x,0)
oo
h[3] >>> limit(((1+x**1)**(1/x)),x,0)
E
h[4] >>> limit(((1+x**Rational(5,4))**(1/x)),x,0)
1
But when the exponent is > 1 and written in terms of exp or log gruntz
stumbles:
h[1] >>> limit(((1+x**exp(3))**(1/x)),x,0)
oo
h[2] >>> limit(((1+x**log(3))**(1/x)),x,0)
oo
Here are the debug outputs for exponent exp(3) -- fails -- and the rational
approximation of the same:
limitinf((1 + (1/_x)**exp(3))**_x, _x) = oo
+-mrv_leadterm((1 + (1/_x)**exp(3))**_x, _x) = (1, -1)
| +-mrv((1 + (1/_x)**exp(3))**_x, _x) = set([exp(_x*log(1 +
(1/_x)**exp(3)))])
| | +-mrv(exp(_x*log(1 + (1/_x)**exp(3))), _x) = set([exp(_x*log(1 +
(1/_x)**exp(3)))])
| | +-limitinf(_x*log(1 + (1/_x)**exp(3)), _x) = oo
| | | +-mrv_leadterm(_x*log(1 + (1/_x)**exp(3)), _x) = (log(1 +
_x**(-exp(3))), -1)
| | | | +-mrv(_x*log(1 + (1/_x)**exp(3)), _x) = set([_x])
| | | | | +-mrv(_x, _x) = set([_x])
| | | | | +-mrv(log(1 + (1/_x)**exp(3)), _x) = set([_x])
| | | | | +-mrv(1 + (1/_x)**exp(3), _x) = set([_x])
| | | | | +-mrv(1, _x) = set([])
| | | | | +-mrv((1/_x)**exp(3), _x) = set([_x])
| | | | | +-mrv(1/_x, _x) = set([_x])
| | | | | +-mrv(_x, _x) = set([_x])
| | | | +-mrv_leadterm(exp(_x)*log(1 + exp(-_x*exp(3))), _x,
set([exp(_x)])) = (log(1 + exp(-_x*exp(3))), -1)
| | | | +-rewrite(exp(_x)*log(1 + exp(-_x*exp(3))), set([exp(_x)]),
_x, _w)= (log(1 + exp(-_x*exp(3)))/_w, -_x)
| | | | +-sign(_x, _x) = 1
| | | | +-mrv_leadterm(1, _x) = (1, 0)
| | | +-sign(-1, _x) = -1
| | | +-sign(log(1 + _x**(-exp(3))), _x) = 1
| | | +-sign(_x**(-exp(3)), _x) = 1
| | | +-sign(_x, _x) = 1
| | +-mrv(_x*log(1 + (1/_x)**exp(3)), _x) = set([_x])
| | +-mrv(_x, _x) = set([_x])
| | +-mrv(log(1 + (1/_x)**exp(3)), _x) = set([_x])
| | +-mrv(1 + (1/_x)**exp(3), _x) = set([_x])
| | +-mrv(1, _x) = set([])
| | +-mrv((1/_x)**exp(3), _x) = set([_x])
| | +-mrv(1/_x, _x) = set([_x])
| | +-mrv(_x, _x) = set([_x])
| +-rewrite((1 + (1/_x)**exp(3))**_x, set([exp(_x*log(1 +
(1/_x)**exp(3)))]), _x, _w) = (1/_w, -_x*log(1 + (1/_x)**exp(3)))
| +-sign(_x*log(1 + (1/_x)**exp(3)), _x) = 1
| | +-sign(_x, _x) = 1
| | +-sign(log(1 + (1/_x)**exp(3)), _x) = 1
| | +-sign((1/_x)**exp(3), _x) = 1
| | +-sign(1/_x, _x) = 1
| | +-sign(_x, _x) = 1
| +-mrv_leadterm(1, _x) = (1, 0)
+-sign(-1, _x) = -1
+-sign(1, _x) = 1
oo
==========
limitinf((1 + (1/_x)**(1647/82))**_x, _x) = 1
+-mrv_leadterm((1 + (1/_x)**(1647/82))**_x, _x) = (1, 0)
| +-mrv((1 + (1/_x)**(1647/82))**_x, _x) = set([_x])
| | +-mrv(exp(_x*log(1 + (1/_x)**(1647/82))), _x) = set([_x])
| | +-limitinf(_x*log(1 + (1/_x)**(1647/82)), _x) = 0
| | | +-mrv_leadterm(_x*log(1 + (1/_x)**(1647/82)), _x) = (0, 0)
| | | | +-mrv(_x*log(1 + (1/_x)**(1647/82)), _x) = set([_x])
| | | | | +-mrv(_x, _x) = set([_x])
| | | | | +-mrv(log(1 + (1/_x)**(1647/82)), _x) = set([_x])
| | | | | +-mrv(1 + (1/_x)**(1647/82), _x) = set([_x])
| | | | | +-mrv(1, _x) = set([])
| | | | | +-mrv((1/_x)**(1647/82), _x) = set([_x])
| | | | | +-mrv(1/_x, _x) = set([_x])
| | | | | +-mrv(_x, _x) = set([_x])
| | | | +-mrv_leadterm(exp(_x)*log(1 + exp(-1647*_x/82)), _x,
set([exp(_x)]))= (0, 0)
| | | | +-rewrite(exp(_x)*log(1 + exp(-1647*_x/82)),
set([exp(_x)]), _x, _w) = (log(1 + _w**(1647/82))/_w, -_x)
| | | | | +-sign(_x, _x) = 1
| | | | | +-mrv_leadterm(1, _x) = (1, 0)
| | | | +-limitinf(log(_x**(-2))/log((1/_x)**(1647/82)), _x) =
164/1647
| | | | | +-mrv_leadterm(-log(_x**2)/log((1/_x)**(1647/82)), _x) =
(164/1647, 0)
| | | | | | +-mrv(-log(_x**2)/log((1/_x)**(1647/82)), _x) =
set([_x])
| | | | | | | +-mrv(-1, _x) = set([])
| | | | | | | +-mrv(log(_x**2)/log((1/_x)**(1647/82)), _x) =
set([_x])
| | | | | | | +-mrv(1/log((1/_x)**(1647/82)), _x) = set([_x])
| | | | | | | | +-mrv(log((1/_x)**(1647/82)), _x) = set([_x])
| | | | | | | | +-mrv((1/_x)**(1647/82), _x) = set([_x])
| | | | | | | | +-mrv(1/_x, _x) = set([_x])
| | | | | | | | +-mrv(_x, _x) = set([_x])
| | | | | | | +-mrv(log(_x**2), _x) = set([_x])
| | | | | | | +-mrv(_x**2, _x) = set([_x])
| | | | | | | +-mrv(_x, _x) = set([_x])
| | | | | | +-mrv_leadterm(164/1647, _x, set([exp(_x)])) =
(164/1647, 0)
| | | | | +-sign(0, _x) = 0
| | | | | +-limitinf(164/1647, _x) = 164/1647
| | | | +-limitinf(log(_x**(-4))/log((1/_x)**(1647/82)), _x) =
328/1647
| | | | | +-mrv_leadterm(-log(_x**4)/log((1/_x)**(1647/82)), _x) =
(328/1647, 0)
| | | | | | +-mrv(-log(_x**4)/log((1/_x)**(1647/82)), _x) =
set([_x])
| | | | | | | +-mrv(-1, _x) = set([])
| | | | | | | +-mrv(log(_x**4)/log((1/_x)**(1647/82)), _x) =
set([_x])
| | | | | | | +-mrv(1/log((1/_x)**(1647/82)), _x) = set([_x])
| | | | | | | | +-mrv(log((1/_x)**(1647/82)), _x) = set([_x])
| | | | | | | | +-mrv((1/_x)**(1647/82), _x) = set([_x])
| | | | | | | | +-mrv(1/_x, _x) = set([_x])
| | | | | | | | +-mrv(_x, _x) = set([_x])
| | | | | | | +-mrv(log(_x**4), _x) = set([_x])
| | | | | | | +-mrv(_x**4, _x) = set([_x])
| | | | | | | +-mrv(_x, _x) = set([_x])
| | | | | | +-mrv_leadterm(328/1647, _x, set([exp(_x)])) =
(328/1647, 0)
| | | | | +-sign(0, _x) = 0
| | | | | +-limitinf(328/1647, _x) = 328/1647
| | | | +-limitinf(log(_x**(-6))/log((1/_x)**(1647/82)), _x) =
164/549
| | | | | +-mrv_leadterm(-log(_x**6)/log((1/_x)**(1647/82)), _x) =
(164/549, 0)
| | | | | | +-mrv(-log(_x**6)/log((1/_x)**(1647/82)), _x) =
set([_x])
| | | | | | | +-mrv(-1, _x) = set([])
| | | | | | | +-mrv(log(_x**6)/log((1/_x)**(1647/82)), _x) =
set([_x])
| | | | | | | +-mrv(1/log((1/_x)**(1647/82)), _x) = set([_x])
| | | | | | | | +-mrv(log((1/_x)**(1647/82)), _x) = set([_x])
| | | | | | | | +-mrv((1/_x)**(1647/82), _x) = set([_x])
| | | | | | | | +-mrv(1/_x, _x) = set([_x])
| | | | | | | | +-mrv(_x, _x) = set([_x])
| | | | | | | +-mrv(log(_x**6), _x) = set([_x])
| | | | | | | +-mrv(_x**6, _x) = set([_x])
| | | | | | | +-mrv(_x, _x) = set([_x])
| | | | | | +-mrv_leadterm(164/549, _x, set([exp(_x)])) = (164/549,
0)
| | | | | +-sign(0, _x) = 0
| | | | | +-limitinf(164/549, _x) = 164/549
| | | | +-limitinf(log(_x**(-8))/log((1/_x)**(1647/82)), _x) =
656/1647
| | | | +-mrv_leadterm(-log(_x**8)/log((1/_x)**(1647/82)), _x) =
(656/1647, 0)
| | | | | +-mrv(-log(_x**8)/log((1/_x)**(1647/82)), _x) =
set([_x])
| | | | | | +-mrv(-1, _x) = set([])
| | | | | | +-mrv(log(_x**8)/log((1/_x)**(1647/82)), _x) =
set([_x])
| | | | | | +-mrv(1/log((1/_x)**(1647/82)), _x) = set([_x])
| | | | | | | +-mrv(log((1/_x)**(1647/82)), _x) = set([_x])
| | | | | | | +-mrv((1/_x)**(1647/82), _x) = set([_x])
| | | | | | | +-mrv(1/_x, _x) = set([_x])
| | | | | | | +-mrv(_x, _x) = set([_x])
| | | | | | +-mrv(log(_x**8), _x) = set([_x])
| | | | | | +-mrv(_x**8, _x) = set([_x])
| | | | | | +-mrv(_x, _x) = set([_x])
| | | | | +-mrv_leadterm(656/1647, _x, set([exp(_x)])) =
(656/1647, 0)
| | | | +-sign(0, _x) = 0
| | | | +-limitinf(656/1647, _x) = 656/1647
| | | +-sign(0, _x) = 0
| | | +-limitinf(0, _x) = 0
| | +-mrv(_x*log(1 + (1/_x)**(1647/82)), _x) = set([_x])
| | +-mrv(_x, _x) = set([_x])
| | +-mrv(log(1 + (1/_x)**(1647/82)), _x) = set([_x])
| | +-mrv(1 + (1/_x)**(1647/82), _x) = set([_x])
| | +-mrv(1, _x) = set([])
| | +-mrv((1/_x)**(1647/82), _x) = set([_x])
| | +-mrv(1/_x, _x) = set([_x])
| | +-mrv(_x, _x) = set([_x])
| +-mrv_leadterm((1 + exp(-1647*_x/82))**exp(_x), _x, set([exp(_x)])) =
(1, 0)
| +-rewrite((1 + exp(-1647*_x/82))**exp(_x), set([exp(_x)]), _x, _w)
= ((1 + _w**(1647/82))**(1/_w), -_x)
| | +-sign(_x, _x) = 1
| | +-mrv_leadterm(1, _x) = (1, 0)
| +-limitinf(log(_x**(-2))/log((1/_x)**(1647/82)), _x) = 164/1647
| +-mrv_leadterm(-log(_x**2)/log((1/_x)**(1647/82)), _x) =
(164/1647, 0)
| | +-mrv(-log(_x**2)/log((1/_x)**(1647/82)), _x) = set([_x])
| | | +-mrv(-1, _x) = set([])
| | | +-mrv(log(_x**2)/log((1/_x)**(1647/82)), _x) = set([_x])
| | | +-mrv(1/log((1/_x)**(1647/82)), _x) = set([_x])
| | | | +-mrv(log((1/_x)**(1647/82)), _x) = set([_x])
| | | | +-mrv((1/_x)**(1647/82), _x) = set([_x])
| | | | +-mrv(1/_x, _x) = set([_x])
| | | | +-mrv(_x, _x) = set([_x])
| | | +-mrv(log(_x**2), _x) = set([_x])
| | | +-mrv(_x**2, _x) = set([_x])
| | | +-mrv(_x, _x) = set([_x])
| | +-mrv_leadterm(164/1647, _x, set([exp(_x)])) = (164/1647, 0)
| +-sign(0, _x) = 0
| +-limitinf(164/1647, _x) = 164/1647
+-sign(0, _x) = 0
+-limitinf(1, _x) = 1
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