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I'm searching to perform this multivariate limit (correctly)

asked 2020-09-05 04:14:41 -0500

Teo7 gravatar image

I'm searching to perform this kind of limit (without restricting and executing the limit to a variable):

$$ \lim_{(x, y)\to(0, 0)}\frac{x^3y}{x^6+y^2} $$ In the documentation I didn't find a multivariate limit function..

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answered 2020-09-05 09:33:24 -0500

Emmanuel Charpentier gravatar image

This smells of homework, so I'll just submit a simple suggestion : you may find inspiration (but not necessarily a solution) in this book (pp.91-2).

Hoping that this may incite you to read the rest of this (excellen)t book...

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OK thanks. The problem is that I know how to perform this type of integrals, I just need a tool that gives me confirmation of the goodness of the results

Teo7 gravatar imageTeo7 ( 2020-09-05 11:39:17 -0500 )edit

The problems are :

  • As far as I can tell, you didn't say a word about an integral ; and

  • while Sage will happily tell you :

    sage: var("y") y sage: (x^3*y/(x^6+y^2)).limit(y=0).limit(x=0) 0

This is a result of a "mechanical" computation, with no attempt to prove the its validity. Sage may however give you a hint:

sage: var("x,y,r,theta")
(x, y, r, theta)
sage: f(x,y)=x^3*y/(x^6+y^2)
sage: g(r,theta)=f(x,y).subs({x:r*cos(theta),y:r*sin(theta)})
sage: parametric_plot3d([r*cos(theta),r*sin(theta),g(r,theta)],(r,0,1/20),(theta,0,2*pi))

What's this "hole" ? (Hint : not as easy as analysing as it seems...).

Computing isn't enough : you have to think...

Emmanuel Charpentier gravatar imageEmmanuel Charpentier ( 2020-09-05 15:22:56 -0500 )edit

Got it, you've been very helpful. ps I wanted to write "limits", but I was solving integrals at that time ...

Teo7 gravatar imageTeo7 ( 2020-09-05 19:36:38 -0500 )edit

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Asked: 2020-09-05 04:14:41 -0500

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Last updated: Sep 05