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Re: Limit and Root Objects

• To: mathgroup at smc.vnet.net
• Subject: [mg72296] Re: Limit and Root Objects
• From: dh <dh at metrohm.ch>
• Date: Tue, 19 Dec 2006 06:35:09 -0500 (EST)
• References: <em606c$2o1$1@smc.vnet.net>

Hi Andrzej,

It is definitly a bug. The reason for the bug may be that not only the 

function is not continuous at b == -(3/2^(2/3)), but also the numbering 

changes. There are three different real roots for b<-(3/2^(2/3)), 

therefore, the first is the smallest. For b=-(3/2^(2/3)) the two lowest 

roots merge and for b>-(3/2^(2/3)) the two   "former" lowest become 

complex. And now the first root is the "former" highest. Mathematica seems to 

keep the number of the root in the limit process. To make the bug even 

worse, the wrong first root at b=-(3/2^(2/3)) is a double root and 

therefore, reduced to a quadratic root object.

Daniel





Andrzej Kozlowski wrote:

> It is easy to check that the function

> 

> 

> f[b_] := Root[#1^3 + b*#1 - 1 & , 1]

> 

> is discontinuous at b, where

> 

> 

> Reduce[Resultant[x^3 + b*x - 1, D[x^3 + b*x - 1, x], x] == 0, b, Reals]

> 

> b == -(3/2^(2/3))

> 

> indeed this was not so long ago discussed in connection with  a  

> little argument about "usefulness' of Root objects. In view of this,  

> isn't the following a bug?

> 

> 

> u = Limit[f[b], b -> -(3/2^(2/3)), Direction -> 1]

> 

> 

> Root[2*#1^3 + 1 & , 1]

> 

> 

> v = Limit[Root[#1^3 + b*#1 - 1 & , 1], b -> -(3/2^(2/3)), Direction - 

>  > -1]

> 

> 

> Root[2*#1^3 + 1 & , 1]

> 

> 

> u == v

> 

> True

> 

> It looks like Limit is making life too easy for itself by assuming  

> continuity.

> 

> Using NLimit shows that things are not as simple:

> 

> 

> w = NLimit[f[b], b -> -(3/2^(2/3)), Direction -> -1]

> 

> 1.5874010343874532

> 

> 

> z = NLimit[Root[#1^3 + b*#1 - 1 & , 1], b -> -(3/2^(2/3)), Direction - 

>  > 1]

> 

> 

> -0.7937180869283765

> 

> Andrzej Kozlowski

>