Re: Limit of an expression?
- To: mathgroup at smc.vnet.net
- Subject: [mg67596] Re: Limit of an expression?
- From: Jean-Marc Gulliet <jeanmarc.gulliet at gmail.com>
- Date: Sat, 1 Jul 2006 05:12:52 -0400 (EDT)
- Organization: The Open University, Milton Keynes, UK
- References: <200606280751.DAA03399@smc.vnet.net> <e7vkut$smg$1@smc.vnet.net> <e82omj$rau$1@smc.vnet.net>
- Sender: owner-wri-mathgroup at wolfram.com
David W.Cantrell wrote:
> 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,
David,
Please find hereunder a "font safe" version of Andrzej's post.
In[1]:=
(Limit[(1 - E^((-mu)*t*(s - 1 - lambda/mu)))/(s - 1 - lambda/mu), t ->
Infinity,
Assumptions -> {mu > 0 && #1[s, 1 + lambda/mu]}] & ) /@ {Greater,
Equal, Less}
Out[1]=
mu
{-(------------------), 0, Infinity}
lambda + mu - mu s
> 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
>
You are right.
> which does not seem to be reasonable in Mathematica. I would have expected
> Indeterminate instead.
I did not notice that at first (with the lambda, mu things :-) , but I
agree.
Best regards,
Jean-Marc