RE: limits

On March 25, 2007 11:54 AM Ondrej Certik wrote:
> 
> I am sorry for this stupid question - but how can I
> calculate the limit of:
> 
> (3**x+5**x)**(1/x)
> 
> for x->+infinity?

Sorry, that seems like an intelligent question to me. ;)
But really there are no stupid questions, just stupid answers.

In fact I think it is a bug or at least a deficiency in Axiom
so I have reported it to IssueTracker here:

http://wiki.axiom-developer.org/345

Thanks for pointing this out. Perhaps this can be corrected
fairly easily.

> The result is 5 as you can check by hand:
> 
> (3**x+5**x)**(1/x) = 5 * ((3/5)**x+1)**(1/x) -> 5
> 
> When I tried that in axiom:
> 
> limit((3**x+5**x)**(1/x), x=%+infinity)

In Axiom you should write:

 limit((3**x+5**x)**(1/x), x=%plusInfinity)

But if you got "failed" as a result you probably wrote
%plusInfinity and it's just a typo in your email.

> 
> or the equivalent problem:
> 
> limit((3**(1/x)+5**(1/x))**(x), x=0, "right")
> 
> I got "failed".
> 

Same bug I guess.

> Also, this works correctly:
> 
> simplify(2**x * 2**(2*x))
> 
> simplifies to 2**(3x)
> 
> but this doesn't simplify at all:
> 
> simplify(2**(5*x)/2**(4*x))
> 

In general, expression simplification in Axiom is not as
powerful as in most other CAS and often needs a little help.
Fortunately Axiom does have a fairly simple powerful pattern
matching feature so one way these simplifications can be done
is as follows:

(1) -> reduceExp := rule exp(x*log(y))==y^x

          x log(y)     x
   (1)  %e         == y
          Type: RewriteRule(Integer,Integer,Expression Integer)

(2) -> reduceExp  normalize (2**x * 2**(2*x))

         x
   (2)  8
                                       Type: Expression Integer

(3) -> reduceExp  normalize (2**(5*x)/2**(4*x))

         x
   (3)  2
                                       Type: Expression Integer

Regards,
Bill Page.