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MathGroup Archive 2006

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Re: Problem with Limit

  • To: mathgroup at smc.vnet.net
  • Subject: [mg66958] Re: Problem with Limit
  • From: ab_def at prontomail.com
  • Date: Mon, 5 Jun 2006 03:48:09 -0400 (EDT)
  • References: <200605311030.GAA13862@smc.vnet.net><e5mh78$kaa$1@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

Andrzej Kozlowski wrote:
> On 31 May 2006, at 19:30, Tony Harker wrote:
>
> >
> > Can anybody explain why, for
> >
> > t =Sqrt[(k*m1 + k*m2 + m1*g + m2*g -
> >          (1/2)*Sqrt[4*(m1 + m2)^2*(k + g)^2 - 16*k*m1*m2*(k + 2*g)])/
> >         (m1*m2)]/(1 + (k*m1 - k*m2 + m1*g - m2*g +
> >           (1/2)*Sqrt[4*(m1 + m2)^2*(k + g)^2 - 16*k*m1*m2*(k +
> > 2*g)])^2/
> >         (4*m1*m2*g^2)) + Sqrt[(k*m1 + k*m2 + m1*g + m2*g +
> >          (1/2)*Sqrt[4*(m1 + m2)^2*(k + g)^2 - 16*k*m1*m2*(k + 2*g)])/
> >         (m1*m2)]/(1 + ((-k)*m1 + k*m2 - m1*g + m2*g +
> >           (1/2)*Sqrt[4*(m1 + m2)^2*(k + g)^2 - 16*k*m1*m2*(k +
> > 2*g)])^2/
> >         (4*m1*m2*g^2));
> >
> > Limit[t, g -> 0]
> > Limit[Together[t], g -> 0]
> >
> > both give zero, whereas
> >
> > Limit[Simplify[Together[t]], g -> 0]
> >
> > gives a non-zero result? This is with Mathe4matica 5.2.
> >
> >
> > Dr A.H. Harker
> > Department of Physics and Astronomy
> > University College London
> > Gower Street
> > London
> > WC1E 6BT
> >
> >
>
>
> Mathematica 5.1 gives 0 in all of the above cases. Moreover:
>
> In[48]:=
> t + O[g]^1
>
> Out[48]=
> O[g]^1
>
> which agrees with it.
>
> Andrzej Kozlowski
> Tokyo, Japan

The problem is that at some stage in the computation we get a product
like (a - Sqrt[a^2])*(a + Sqrt[a^2]). The first 'non-zero' term in the
series for t will contain a hidden infinity:

In[2]:= t + O[g]^3 // Simplify

Out[2]= SeriesData[g, 0, {ComplexInfinity}, 2, 3, 1]

A simpler example is

1/(A + (a + Sqrt[a^2])/b) + 1/(B + (a - Sqrt[a^2])/b) + O[b]^2

A useful trick is to take the series expansion of 1/t, let the zeros
(hopefully) cancel, and then take the reciprocal of the series:

In[3]:= 1/(1/t + O[g] // Simplify)

Out[3]= SeriesData[g, 0, {-(m2*(Sqrt[k^2*(m1 -
m2)^2]*(Sqrt[(-Sqrt[k^2*(m1 - m2)^2] + k*(m1 + m2))/(m1*m2)] -
Sqrt[(Sqrt[k^2*(m1 - m2)^2] + k*(m1 + m2))/(m1*m2)]) - k*(m1 -
m2)*(Sqrt[(-Sqrt[k^2*(m1 - m2)^2] + k*(m1 + m2))/(m1*m2)] +
Sqrt[(Sqrt[k^2*(m1 - m2)^2] + k*(m1 + m2))/(m1*m2)]))^3)/(16*k^4*(m1 -
m2)^3)}, 0, 1, 1]

Maxim Rytin
m.r at inbox.ru


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