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MathGroup Archive 2006

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Re: Limit of an expression?

  • To: mathgroup at smc.vnet.net
  • Subject: [mg67520] Re: [mg67479] Limit of an expression?
  • From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
  • Date: Thu, 29 Jun 2006 00:09:54 -0400 (EDT)
  • References: <200606280751.DAA03399@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

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}

Andrzej Kozlowski
Tokyo, Japan


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