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Re: Limit and Root Objects


On 18 Dec 2006, at 17:35, 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


I forgot to add that the correct value of

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

is  Root[#1^3 - 4 & , 1]  (and not Root[2*#1^3 + 1 & , 1]).


Andrzej Kozlowski 


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