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Re: More about l`Hopital`s rule
 To: mathgroup at smc.vnet.net
 Subject: [mg24571] Re: [mg24560] More about l`Hopital`s rule
 From: Andrzej Kozlowski <andrzej at tuins.ac.jp>
 Date: Tue, 25 Jul 2000 00:56:04 0400 (EDT)
 Sender: ownerwrimathgroup at wolfram.com
on 7/24/00 9:04 AM, Joaquín González de Echavarri at
jge at clientes.euskaltel.es wrote:
> Different answers:
>
> In= Limit[(a^2  a*x)/(a  Sqrt[a*x]), x > a]
>
> Out= 0
>
> In=Limit[Simplify[(a^2  a*x)/(a  Sqrt[a*x])], x > a]
>
> Out= 0
>
> In= Limit[FullSimplify[(a^2  a*x)/(a  Sqrt[a*x])], x > a]
>
> 2
> Out= a + Sqrt[a ]
>
> Wrong answer:
>
> In= Limit[(Sqrt[x + a^2]  a)/(Sqrt[x + b^2]  b), x > 0]
>
> 2
> a  Sqrt[a ]
> Out= 
> 2
> b  Sqrt[b ]
>
> The right answer is b/a.
>
> Any suggestion?
>
> Joako
>
This is basically the same situation as in the original problem in the
l'Hospital rule thread. You really should have read posted answers to that
thread before sending your own.
The problem is that you are quite wrong when you write:
> The right answer is b/a.
You could easily check yourself that this is not always so, e.g.
In[1]:=
f[x_] := (Sqrt[a^2 + x]  a)/(Sqrt[b^2 + x]  b)
In[2]:=
a = 6; b = 3;
In[3]:=
f[0.0001]
Out[3]=
2.
or
In[4]:=
Limit[f[x], x > 0]
Out[4]=
2
This is a/b not b/a!
On the other hand:
In[5]:=
a = 6; b = 3;
In[6]:=
Limit[f[x], x > 0]
Out[6]=
1

2
The point is that Mathematica simply chose a differnet branch branch of the
multivalued solution set and you are chosing a different one. If
Mathematica's answer is "wrong" then so is yours. In fact, Mathematica's
answer covers more cases than your won.
One can find all the answers using Mathematica if one approaches the problem
a little more carefuly:
In[1]:=
h[x_] := Sqrt[a^2 + x]  a + O[x]^2;
g[x_] := Sqrt[b^2 + x]  b + O[x]^2;
In[2]:=
conds = Partition[
Flatten[NestList[RotateLeft,
Outer[Apply, {Less, Greater}, {{a, 0}, {b, 0}}, 1] // Flatten, 1]],
2]
Out[2]=
{{a < 0, b < 0}, {a > 0, b > 0}, {b < 0, a > 0}, {b > 0, a < 0}}
In[59]:=
Limit[Simplify[Normal[h[x]]/Normal[g[x]], #], x > 0] & /@ conds
Out[59]=
a b
{, , 0, (Infinity Sign[a b])}
b a
This last element in this list should be , of course, just Infinity.

Andrzej Kozlowski
Toyama International University, JAPAN
For Mathematica related links and resources try:
<http://www.sstreams.com/Mathematica/>
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