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Limit and Root Objects

• To: mathgroup at smc.vnet.net
• Subject: [mg72284] Limit and Root Objects
• From: Andrzej Kozlowski <andrzej at akikoz.net>
• Date: Mon, 18 Dec 2006 06:55:56 -0500 (EST)

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