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Re: Limit of an expression?

  • To: mathgroup at smc.vnet.net
  • Subject: [mg67625] Re: Limit of an expression?
  • From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
  • Date: Sun, 2 Jul 2006 06:28:23 -0400 (EDT)
  • Sender: owner-wri-mathgroup at wolfram.com


On 30 Jun 2006, at 11:49, David W. Cantrell wrote:


> [Message also posted to: comp.soft-sys.math.mathematica]
>
> Andrzej Kozlowski <akoz at mimuw.edu.pl> wrote:
>
>> On 28 Jun 2006, at 16:51, Virgil Stokes wrote:
>>
>>
>>> In the following expression, s is an integer (>= 1), Lambda, Mu,  
>>> and t
>>> are real numbers and all > 0.
>>> What is the limit of the following as t goes to infinity?
>>>
>>> \!\(\(1 - \[ExponentialE]\^\(\(-\[Mu]\)\ t\ \((s - 1 - \
>>> \[Lambda]\/\[Mu])\)\)\)\/\(s - 1 - \[Lambda]\/\[Mu]\)\)
>>>
>>> --V. Stokes
>>>
>>>
>>
>> Unless you made a mistake in the formula you posted,  the answer
>> depends  on the sign of s - 1 -


>> λ/μ. Mathematica can deal with all
>> three possible cases (it is also pretty obvious when done by hand):
>>
>> (Limit[(1 - E^((-μ)*t*
>>          (s - 1 - λ/μ)))/
>>       (s - 1 - λ/μ),
>>      t -> Infinity,
>>      Assumptions ->
>>       {μ > 0 && #1[s,
>>          1 + λ/μ]}] & ) /@
>>    {Greater, Equal, Less}
>>
>> {-(μ/(λ - s*μ + μ)),
>>    0, Infinity}
>>
>
> Much of the above is illegible to me, but I'm guessing that the  
> middle case
> is equivalent to
>
> In[1]:= Assuming[a==0, Limit[(1 - Exp[a t])/a, t->Infinity]]
>
> Out[1]= 0
>
> which does not seem to be reasonable in Mathematica. I would have  
> expected
> Indeterminate instead.
>
> David
>


I agree. (I neglected to pay careful attention to the output when I  
posted the above - sometimes I have too much trust in Mathematica I  
guess ;-) ). If you look at Trace with the option TraceInternal->True  
you can see how Mathematica arrives at this. Essentially, at some  
point it evaluates


Simplify[1 - E^(a*t), Assumptions -> {a == 0}]

and then it concludes that the whole expression is 0, without ever  
noticing that the denominator is also assumed to be zero.

This, I think, should count as a bug.


Andrzej Kozlowski
Tokyo, Japan


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