Re: Limit of an expression?
- To: mathgroup at smc.vnet.net
- Subject: [mg67578] Re: Limit of an expression?
- From: Jean-Marc Gulliet <jeanmarc.gulliet at gmail.com>
- Date: Sat, 1 Jul 2006 05:12:09 -0400 (EDT)
- Organization: The Open University, Milton Keynes, UK
- References: <e7td7i$3lg$1@smc.vnet.net>
- Sender: owner-wri-mathgroup at wolfram.com
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 > ===== Sorry for the mess with the fonts in my previous post. Please, find hereunder a readable version of it. ===== Hi Virgil, Contrary to some functions such as Integrate, Limit does not returned conditional answers; that is, Limit will return a solution only if it can establish the convergence or divergence of the expression without taking in account the values of parameters. For instance, compare the following answers to a simple limit In[1]:= Limit[E^(-t), t -> Infinity] Out[1]= 0 In[2]:= Limit[E^((-a)*t), t -> Infinity] Out[2]= -a t Limit[E , t -> Infinity] Of course, in the latter case you can add an assumption on the parameter 'a' and get the answer In[3]:= Limit[E^((-a)*t), t -> Infinity, Assumptions -> a > 0] Out[3]= 0 Now, let's go back to your original question. In Mathematica form, we have In[4]:= Limit[(1 - E^((-mu)*t*(s - 1 - lambda/mu)))/ (s - 1 - lambda/mu), t -> Infinity, Assumptions -> {Element[s, Integers], s > 0, lambda > 0, mu > 0, t > 0}] Out[4]= -mu (-1 - lambda/mu + s) t 1 - E Limit[-------------------------------, t -> Infinity, lambda -1 - ------ + s mu Assumptions -> {Element[s, Integers], s > 0, lambda > 0, mu > 0, t > 0}] As stated above, the expression is returned unevaluated since there is indeed three possible answers depending on the respective value of s, mu, and lambda. We can see by inspection that the limit of the exponential function depends on the value of In[5]:= Simplify[E^((-mu)*t*(s - 1 - lambda/mu)), Assumptions -> {Element[s, Integers], s > 0, lambda > 0, mu > 0, t > 0}] Out[5]= (lambda + mu - mu s) t E Therefore, we compute the different values in three steps In[6]= Limit[(1 - E^((-mu)*t*(s - 1 - lambda/mu)))/ (s - 1 - lambda/mu), t -> Infinity, Assumptions -> {Element[s,Integers], s > 0, lambda > 0, mu > 0, t > 0, lambda + mu - mu*s > 0}] Out[6]= Infinity In[7]:= Limit[(1 - E^((-mu)*t*(s - 1 - lambda/mu)))/ (s - 1 - lambda/mu), t -> Infinity, Assumptions -> {Element[s, Integers], s > 0, lambda > 0, mu > 0, t > 0, lambda + mu - mu*s == 0}] Out[7]= 0 In[8]:= Limit[(1 - E^((-mu)*t*(s - 1 - lambda/mu)))/ (s - 1 - lambda/mu), t -> Infinity, Assumptions -> {Element[s, Integers], s > 0, lambda > 0, mu > 0, t > 0, lambda + mu - mu*s < 0}] Out[8]= mu -(------------------) lambda + mu - mu s Best regards, Jean-Marc