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