Solving the Limit of a Two-Variable Function

[transphenomen]

This question came up in a book I am using for self study. I was going to just skip the question but then I remembered I can ask for help here.

Homework Statement

Let a,b,c,d be nonnegative integers. For what values of a,b,c,d does
<br />
<br /> \lim_{\substack{x\rightarrow 0\\y\rightarrow 0}} \frac{x^a y^b}{x^{2c}+y^{2d}}<br />
exist? For those values, what is the limit?(ignore the set A if you can see it, I think its a glitch)

Homework Equations

Just basic definitions of limits. Nothing beyond a first semester of real analysis should be needed.

The Attempt at a Solution

When I assumed that a=b=c=d=1 and fix x=0, then the limit is 0, yet if I assumed x=y then the limit was 1/2 and determined that the limit doesn't exist. However, I do not know how to generalize it to any value of a,b,c,d. I was thinking of just assuming its a 6 dimensional problem, but there are problems with doing that so I came here for help.

 

[SammyS]

Try using polar coordinates.

 

[transphenomen]

Try using polar coordinates.

Well, if a=b=c=d=1, I get a neat little tidbit saying that the limit as r goes to zero is sin(theta)cos(theta) for what ever theta you travel through to get to 0. Though this confirms my assumption that the limit doesn't exist for when a=b=c=d=1, it doesn't tell me anything about for when a,b,c,d are independent of each other. When a,b,c,d are different from each other, I start getting powers of r in the numerator and denominator which doesn't help much. I suppose I could be missing something, so if so, could I get a better hint?

 

[transphenomen]

I had an idea and maybe you can tell me if I am right. Since sin and cos are between -1 and 1, and the denominators values of sin and cos is always greater than or equal to 0 then if a + b > max{2c,2d}, then the limit goes to 0, if not then the limit doesn't exist. Am I on the right track?

 

[SammyS]

Assume that c ≥ d and see what you get in that case. This let's you factor out r^2d in the denominator.

Added in Edit:

Try for instance: a=1, b=2, c=3, d=2 .