Re: Limit and Root Objects

*To*: mathgroup at smc.vnet.net*Subject*: [mg72285] Re: Limit and Root Objects*From*: Andrzej Kozlowski <akoz at mimuw.edu.pl>*Date*: Mon, 18 Dec 2006 06:56:00 -0500 (EST)*References*: <B6993FE7-D191-4A9D-9741-C4CA161BA14A@akikoz.net>

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