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Unexpected limit result from Wolfram|Alpha?

Posted 7 days ago
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lim (x,y)->(0,0) 2xy^2/(x^4+y^2) is not D.N.E, it should goes to 0

7 Replies
Posted 6 days ago

The result is indeed 0. Maybe you defined the formula incorrectly, you have to define it like this

Posted 6 days ago

The result from WolframAlpha includes

(limit does not exist) (value may depend on x, y path in complex space)

Consider the path from x == 1, y == -Sqrt[-1] to x == 0, y == 0 where points on that path maintain the denominator == 0, I believe that limit will be ComplexInfinity

Compare that with paths where both x and y are Real. I believe those limits will be zero.

I think you can approximate any (reasonable) path in the vicinity of { 0, 0 } by a straight line . So look at this, which is independent of a and b

Limit[2 x y^2/(x^4 + y^2) /. {x -> a t, y -> b t}, t -> 0]

Also the following limit is independent of a and b:

Limit[2 x^2 y/(x^4 + y^2) /. {x -> a t, y -> b t}, t -> 0]

but the function 2 x^2 y/(x^4 + y^2) has no limit at the origin, as we can see by setting y->a x^2.

Back to the original function 2 x y^2/(x^4 + y^2), it is dominated by 2Norm[{x,y}]:

Reduce[RealAbs[(2 x y^2)/(x^4 + y^2)] <= 2 Norm[{x, y}], Reals]

and this proves that its limit is zero at the origin, if we work in the real domain.

but the function 2 x^2 y/(x^4 + y^2) has no limit at the origin,

Ok. But where is my error?

2 x^2 y/(x^4 + y^2) /. x -> 0 // FullSimplify
2 x^2 y/(x^4 + y^2) /. y -> 0 // FullSimplify
2 x^2 y/(x^4 + y^2) /. x -> y // FullSimplify
2 x^2 y/(x^4 + y^2) /. y -> x // FullSimplify
2 x^2 y/(x^4 + y^2) /. y -> a x^2 // FullSimplify

Along all straight lines the limits are the same, but the limits are not uniform. You can make a plot:

Plot3D[2 x^2 y/(x^4 + y^2), {x, -1, 1}, {y, -1, 1},
 PlotPoints -> 200, Exclusions -> Automatic,
 MeshFunctions -> {#3 &}]

In every neighbourhood of the origin the function oscillates between the values 1 and -1. It does not converge to zero. Still, along each straight line y == a x the function goes to zero, although it is not obvious from the picture:

Plot3D[2 x^2 y/(x^4 + y^2), {x, -1, 1}, {y, -1, 1},
 PlotPoints -> 200, Exclusions -> Automatic,
 MeshFunctions -> {#2/#1 &},
 Mesh -> {Tan[Pi/2 Range[-19, 19]/20]}]

Maybe the wireframe version is clearer:

Plot3D[2 x^2 y/(x^4 + y^2), {x, -1, 1}, {y, -1, 1},
 PlotPoints -> 200, Exclusions -> Automatic,
 MeshFunctions -> {#2/#1 &},
 Mesh -> {Tan[Pi/2 Range[-19, 19]/20]}, PlotStyle -> None]
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