Hello everybody!
Does anybody know why WolframAlpha gives the answer "0" to
limit[(x*y^2)/(x^2 + y^4),{x,y}->{0,0}]
against the fact that, although on many paths the limit is 0, in each neighborhood of the origin there are:
an infinite number of points (of the kind x = y^2) on which the function simplifies to the constant 1/2
an infinite number of points (of the kind x = - y^2) on which the function simplifies to the constant - 1/2.
In my opinion the right answer should then be "limit does not exist, is path dependent or cannot be determined", (the same WolframAlpha gives, for example, to limit[(x*y)/(x^2 + y^2),{x,y}->{0,0}], which can't be "0" because the function simplifies to the constant 1/2 if you set y = x)
Infact both functions have a discontinuity in the origin, and this can also be shown by Plot3D (using more PlotPoints makes it clearer)
Plot3D[(x y^2)/(x^2 + y^4), {x, -.5, .5}, {y, -.5, .5},
PlotRange -> {-.5, .5}, PlotPoints -> 150]
Plot3D[(x y)/(x^2 + y^2), {x, -.5, .5}, {y, -.5, .5},
PlotRange -> {-.5, .5}]
Am I somehow wrong or is it a bug? Does Mathematica 8, 9 or 10 calculate this kind of limits? (I'm only up to version 7...)
To understand better, I also asked WolframAlpha for
limit[(x^a*y^(a*b))/(x^(2*a) + y^(2*a*b)), {x, y} -> {0, 0}]
with a = 1,2,3,4,5 and b = 1,2,3,4 and still got the answer "0" (except for b = 1) against the fact that the function simplifies to 1/2 on the points where x = y^b (and to the constant -1/2 for x = -y^b if a is odd).
It looks as if the algorithm "checked" only the paths like x = y and not those of the kind x = y^b.
The same fact occurs if you change x and y and ask for
limit[(x^(a*b) y^a)/(x^(2*a*b) + y^(2*a)), {x, y} -> {0, 0}]
Thank you very much, bye!
Mario Gianini