# Multivariable Limit

1. Sep 22, 2011

### Yuqing

1. The problem statement, all variables and given/known data
Find the limit of

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

2. Relevant 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.

3. 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.

2. Sep 22, 2011

### Hootenanny

Staff Emeritus
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.

3. Sep 22, 2011

### Yuqing

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.

4. Sep 22, 2011

### SammyS

Staff Emeritus
Try using polar coordinates.

5. Sep 22, 2011

### SammyS

Staff Emeritus
Use a path given by y = xa .

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