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Re: Same Limit: OK in 5.2, fails in 6.0; Packages gone


jrc wrote:
> Why?
> 
> I have,
> 
> $Assumptions = {a > 0, k1 > 0, k2 >0, x e(in) Reals}
> (repeats ok)
> 
> i1b = (1/a)* Integral from -a/2 to +a/2 of integrand:
> 
>               exp(- i k2 x) exp(i k1 x) dx
> 
> with the expected result,
> 
> i1b = fraction with numerator = 2 Sin[(1/2)a(k1-k2)]
>                  and denominator = a(k1-k2)
> 
> Now I want the limit of this result, as the parameter a goes to infinity:
> 
> Limit[i1b, a -> Infinity]
> 
> Mathematica 5.2 gives correct result, zero;
>    (note this is just lim(sin(x)/x, x -> inf), which is obviously zero)
> 
> Mathematica 6.0 is unable to evaluate the limit.

This is a glitch in Limit handling of $Assumptions. It was written to 
work with an Assumptions option and sometimes messes up the more global 
$Assumptions. Offhand I do not know why it did not also mess up in 
version 5.2.

Workarounds include

(1) Use explicit And (instead of List) for $Assumptions.

$Assumptions = Apply[And,{a>0,k1>0,k2>0,Element[x,Reals]}]

In[10]:= Limit[i1b, a->Infinity]
Out[10]= 0

(2) Use Assumptions option in Limit.

In[12]:= $Assumptions = .

In[13]:= ClearSystemCache[]

In[14]:= Limit[i1b, a->Infinity, 
Assumptions->{a>0,k1>0,k2>0,Element[x,Reals]}]
Out[14]= 0

We'll get the underlying problem fixed.


> Ruskeepaa's "Navigator", 2nd ed, (written for v5.2) claims there
> is a package, "Calculus`Limit`" that makes 'Limit' work better
> (p. 395 and p. 430). However, no such package seems to exist in
> v6.0.

That claim was a stretch. By any reasonable metric that package made 
Limit substantially more error-prone. Which is why the package is gone.


> How many packages no longer exist in 6.0 ???? Is there a list ????
> 
> Can anyone give me a reason for this obvious failure?
> 
> jrc


Daniel Lichtblau
Wolfram Research



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