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Re: Problem with Limit

• To: mathgroup at smc.vnet.net
• Subject: [mg66864] Re: Problem with Limit
• From: "Scout" <Scout at nodomain.com>
• Date: Fri, 2 Jun 2006 04:08:24 -0400 (EDT)
• References: <e5jri2$dos$1@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

"Tony Harker" <a.harker at ucl.ac.uk>  news:e5jri2$dos$1 at smc.vnet.net...
>
> 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
>
>

Could it be a bug of Limit[] command?
However the 0 value of the limit is incorrect.
Limit[] therefore makes no explicit assumptions about functions, so the 
square roots can be the arising problem, specially with more symbols 
(k,g,m1,m2).
Assuming[] doesn't improve it, in fact:

    In[1]:= Assuming[k > 0 && m1 > 0 && m2 > 0, Limit[t, g -> 0]]
    Out[1]= 0

    In[2]:= $Version
    Out[2]= "5.2 for Microsoft Windows (June 20, 2005)"

But,

    In[3]:= v = Limit[Simplify[Together[t]], g->0]

    Out[3]= \!\(\(\(1\/\(2\ k\ \((m1 -
            m2)\)\)\)\((\@\(k\^2\ \((m1 - m2)\)\^2\)\ 
\((\(-\@\(\(\(-\@\(k\^2\
\ \((m1 - m2)\)\^2\)\) + k\ \((m1 + m2)\)\)\/\(m1\ m2\)\)\) + 
\@\(\(\@\(k\^2\ \
\((m1 - m2)\)\^2\) + k\ \((m1 + m2)\)\)\/\(m1\ m2\)\))\) +
      k\ \((m1 -
            m2)\)\ \((\@\(\(\(-\@\(k\^2\ \((m1 - m2)\)\^2\)\) + k\ \((m1 + \
m2)\)\)\/\(m1\ m2\)\) + \@\(\(\@\(k\^2\ \((m1 - m2)\)\^2\) + k\ \((m1 + 
m2)\)\
\)\/\(m1\ m2\)\))\))\)\)\)


    In[4]:= Simplify[v, {m1 > m2 && k > 0}]
    Out[4]= \!\(\@2\ \@\(k\/m2\)\)

Regards,
    ~Scout~