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Re: Problem with Limits; basic calculus

  • To: mathgroup at smc.vnet.net
  • Subject: [mg38934] Re: Problem with Limits; basic calculus
  • From: "David W. Cantrell" <DWCantrell at sigmaxi.org>
  • Date: Wed, 22 Jan 2003 06:09:19 -0500 (EST)
  • References: <b0jfs1$t8q$1@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

sophtwarez at hotmail.com (David Seruyange) wrote:
> Hey all - I'm taking a basic calculus course that uses Mathematica.
> We have been studying limits and I have been using the Limit function
> to check if my answers are correct.
> We were given the following function and asked to determine a limit:
> (3E^(-x) - E^(-3x)) / (E^(-3x) + E^(-x))
>
> Usually the approach is to select the dominant terms, factor and then
> determine the limit.  My initial reason had me select -E^(-3x) in the
> numerator and E^(-3x) in the denominator.  Factoring the terms would
> yield -1, thus the limit for x->infinity.  But I plotted the function
> and the real answer is somewhere near 3.
> I then tried to use the Limit function which is not producing an
> answer (perhaps I'm not sure of the usage).

Your idea of considering the dominant terms is good, but you thought that
the wrong ones were dominant. Since you are working with x -> Infinity,
the dominant terms in numerator and denominator are 3E^(-x) and E^(-x),
resp. Thus, their ratio approaches 3, just as your graph indicated.

I hate to report, however, that your usage of Mathematica was correct.
That its Limit function could not give an answer for such a simple
problem is ludicrous. I had thought that perhaps you just needed to use
the Standard Add-on Package Calculus`Limit`. But I hate to report that
using it is even worse! Rather than merely giving no answer, it then gives
the incorrect answer 0. Even more ludicrous.

There is a simple workaround, however. Just Simplify your function
first, before asking for its limit. Mathematica gives
(-1+3E^(2x))/(1+E^(2x)) as the simplification. Then, using either the
built-in Limit function or the one in the add-on package, the correct
answer, 3, is obtained.

David Cantrell


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