Solving Multivariable Limit: x^2y^2/(x^3+y^3)

[Yuqing]

Homework Statement

Find the limit of

\lim_{(x,y)\rightarrow (0,0)}\frac{x^2y^2}{x^3+y^3}

Homework Equations

I'd like to solve this in a rather elementary manner, so preferably only using the squeeze theorem or through proving the limit doesn't exist via multiple path approach.

The Attempt at a Solution

I've tried substituting y = mxⁿ in general and I've tried bounding the denominator. All to no avail. All paths I've tried so far lead to 0 but I am still not certain that the limit actually exists.

 

[Hootenanny]

All paths I've tried so far lead to 0 but I am still not certain that the limit actually exists.

Indeed, the limit does not exist. When finding the limits of a multivariate function, it is useful to plot the function, this helps you decide on paths of approach.

However, in this case it is useful to note that the denominator changes sign depending on the quadrant, whilst the numerator does not.

 

[Yuqing]

However, in this case it is useful to note that the denominator changes sign depending on the quadrant, whilst the numerator does not.

How exactly would you suggest I approach this? The biggest problem I have is that the numerator is of higher overall order than the denominator. I cannot find a path which does not take me to 0.

 

[SammyS]

Try using polar coordinates.

 

[SammyS]

How exactly would you suggest I approach this? The biggest problem I have is that the numerator is of higher overall order than the denominator. I cannot find a path which does not take me to 0.

Use a path given by y = x^a .

For what power, a, will the orders of the numerator & denominator be equal ?