       • 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: owner-wri-mathgroup at wolfram.com

```on 7/24/00 9:04 AM, Joaquín González de Echavarri at
jge at clientes.euskaltel.es wrote:

>
> 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 ]
>
>
> 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

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:=
f[x_] := (Sqrt[a^2 + x] - a)/(Sqrt[b^2 + x] - b)

In:=
a = -6; b = -3;

In:=
f[0.0001]
Out=
2.

or

In:=
Limit[f[x], x -> 0]
Out=
2

This is a/b not b/a!

On the other hand:

In:=
a = 6; b = 3;

In:=
Limit[f[x], x -> 0]
Out=
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

One can find all the answers using Mathematica if one approaches the problem
a little more carefuly:

In:=
h[x_] := Sqrt[a^2 + x] - a + O[x]^2;
g[x_] := Sqrt[b^2 + x] - b + O[x]^2;

In:=
conds = Partition[
Flatten[NestList[RotateLeft,
Outer[Apply, {Less, Greater}, {{a, 0}, {b, 0}}, 1] // Flatten, 1]],
2]
Out=
{{a < 0, b < 0}, {a > 0, b > 0}, {b < 0, a > 0}, {b > 0, a < 0}}

In:=
Limit[Simplify[Normal[h[x]]/Normal[g[x]], #], x -> 0] & /@ conds
Out=
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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