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

• To: mathgroup at smc.vnet.net
• Subject: [mg78003] Re: [mg77959] Same Limit: OK in 5.2, fails in 6.0; Packages gone
• From: Daniel Lichtblau <danl at wolfram.com>
• Date: Thu, 21 Jun 2007 05:42:23 -0400 (EDT)
• References: <200706200941.FAA10343@smc.vnet.net>

```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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