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

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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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

( 2020-09-05 18:39:17 +0200 )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...

( 2020-09-05 22:22:56 +0200 )edit

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

( 2020-09-06 02:36:38 +0200 )edit