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Unexpected limit result from Wolfram|Alpha?

Posted 10 months 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 10 months ago

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

Posted 10 months 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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