Possibly wrong limit concerning log

[pcworld]

lim(27^log(n,3)/n^3, n=infinity)

returns 0.

However both Wolfram Alpha (http://www.wolframalpha.com/input/?i=lim+n-%3Einfty+27%5Elog3%28n%29%2Fn%5E3) and Maple return 1.

Is this a bug in Sage?

[kcrisman]

lim(27^log(n,3)/n^3, n=infinity)

Indeed, if you use http://en.wikipedia.org/wiki/List_of_logarithmic_identities#Canceling_exponentials you can easily see that the expression equals 1 in the first place! Plotting it yields the same.

But unfortunately Maxima does not seem to have this identity, partly perhaps because it only has the 'natural' logarithm.

However,

(%i7) limit(2^(log(x)/log(2))/x,x,inf);

(%o7)                                  1

so it at least sort of knows this.  I wonder if anyone else has any ideas here?

[ssin...@coe.edu]

On Monday, June 1, 2015 at 7:53:18 PM UTC-5, kcrisman wrote:

 I wonder if anyone else has any ideas here?

sympy, too, finds the limit despite not recognizing the identity:

 

%python

​from sympy import *

x = symbols('x')

expr = 27**(log(x,3)/x**3)

expr, limit(expr, x, oo)

>>> (27**(log(x)/(x**3*log(3))), 1)

Steve

[pcworld]

sympy, too, finds the limit despite not recognizing the identity: 

…

expr = 27**(log(x,3)/x**3)

expr, limit(expr, x, oo)

Your expression has "1/x^3" in its exponent. The expression from the orginal post would be 27**(log(x,3))/x**3, which SymPy correctly simplifies and finds the correct limit for:

>>> simplify(27**(log(x,3))/x**3)

Integer(1)

>>> limit(27**(log(x,3))/x**3,x,oo)

Integer(1)

Other examples of wrong limits with logarithm in exponent in Sage:

lim((((27**(log(n,3))))/n**3),n=infinity) returns 0 (Wolfram Alpha and Maple return 1).

lim((((27**(log(n,3)+1)))/n**3),n=infinity) returns 0 (Wolfram Alpha and Maple return 27).

lim(((27**(log(n,3)+1)-1)/26+n-log(n,3)-1)/n**3,n=infinity) returns 0 (Wolfram Alpha and Maple return 27/26).

Using SageMathVersion6.7, ReleaseDate:2015−05−17.

(SymPy has issues with these limits too, albeit different ones. I've reported them here.)

[kcrisman]

Other examples of wrong limits with logarithm in exponent in Sage:

lim((((27**(log(n,3))))/n**3),n=infinity) returns 0 (Wolfram Alpha and Maple return 1).

lim((((27**(log(n,3)+1)))/n**3),n=infinity) returns 0 (Wolfram Alpha and Maple return 27).

lim(((27**(log(n,3)+1)-1)/26+n-log(n,3)-1)/n**3,n=infinity) returns 0 (Wolfram Alpha and Maple return 27/26).

Using SageMathVersion6.7, ReleaseDate:2015−05−17.

I believe these are all the same Maxima problem. 

https://sourceforge.net/p/maxima/bugs/2972/

[kcrisman]

Update - this is fixed upstream! Though I will point out that the fix involves radcan, which not everyone here is enamored of.