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Let $f(x,y) : \mathbb R^2 \to \mathbb R = \frac {x^a y^b}{x^{2c} + y^{2d}}$ for $a, b,c ,d \in \mathbb N$. For what values of $a, b, c, d$ does $f$ converge as $(x,y) \to (0,0).$

If $a \geq 2c$ or $b \geq 2d$, $f$ converges to $0$ as it approaches the origin. Otherwise, I conjecture that $f$ diverges, but request help proving it.

Since $x^{2c}, y^{2d} \geq 0$, then for $a \geq 2c$ and $|x|,|y| < 1$, $$\left |\frac{x^a y^b}{x^{2c} + y^{2d}} \right | \leq \left | \frac{x^a y^b}{x^{2c}} \right | \leq |y^b| \to 0.$$ A similar argument applies to $b \geq 2d$.

To explore $f$'s behavior when $a < 2c$ and $b < 2d$, I've tried the following:

  1. Observe that $f$ vanishes along the $x$-axis. So to prove divergence, it suffices to find any path where $f$'s limit is not zero.
  2. Consider the behavior of $f$ along particular lines, such as $y = x$. I could not show that it was always above any $\varepsilon$.
  3. Try to relate $\left | \frac{x^{a+b}}{x^{2c} + x^{2d}} \right |$ to $\left |\frac{x^{a+b}}{x^{2c+2d}}\right| \to \infty$, perhaps via partial fraction decomposition. I was not able to make progress here.
  4. Try to find $y$ as a function of $x$ such that $f$ along that path will always be greater than a constant. I tried solving $f(x,y) \geq k$ for $y$ but was unable to.
  5. Write $f$ as $\frac {x^a} {x^{2c}} \cdot \frac {y^b} {1+y^{2d}/x^{2c}}$. The left factor goes to infinity, but the right factor goes to zero, not making this of direct help.
  6. Use L'Hopital's rule on #5. To do this, I need to pick a path which relates $y$ to $x$. The simplest is $y=x$, which gives $\frac 1 {x^{2c - (a+b)} + x^{2d - (a+b)}}$. L'Hopital's rule helps me complete the proof for $2c \geq a + b \land 2d \geq a + b$, but doesn't help when either $2c < a + b$ or $2d < a + b$.
  7. Perhaps my conjecture is wrong, and when either $2c < a + b$ or $2d < a + b$, then $f$ converges. But proving that isn't obvious either.

Can you help me complete the proof?

Update

Please note that I'd like help completing my proof: guidance as to any mistakes I may have made, insights I may have missed, or ways I could take one of my 7 approaches further. (That is, I'm not looking for a give away answer, but help completing my proof.)

SRobertJames
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  • @JohnOmielan Thank you - but I'm looking for help completing my proof. I outlined the 7 approaches I've attempted, and request help taking it further - not a link to a canned solution. – SRobertJames Mar 31 '23 at 01:38
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    A big issue with most of your approaches is the $+$ in the denominator. You'll want a path where $x^{2c}$ and $y^{2d}$ are more "compatible" and the sum can simplify... – aschepler Mar 31 '23 at 02:43
  • @SRobertJames Thank you for clarifying. I've found that most people, when presenting a proof attempt, are mainly interested in getting the result by some means, not necessarily using how they are trying to do it (note that their attempt is fairly often one that will not likely be able to made to work). Since you're mainly interested in completing your proof instead, I've removed my close vote and the comment re: the proposed duplicate of When does the limit $\lim_{(x,y)\to(0,0)} \frac{x^ky^l}{x^{2p}+y^{2q}}$ exist?. – John Omielan Mar 31 '23 at 17:24
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    You might want to look a little harder at aschepler's comment. you don't have to approach $0$ along a line. Any curve $(x(t), y(t))$ will do. Some might make life a lot simpler - say if the two terms in the denominator could be added together when expressed as functions of $t$. – Paul Sinclair Mar 31 '23 at 20:16
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    An alternative approach is to examine the behavior of $\frac 1{|f|}$. – Paul Sinclair Mar 31 '23 at 20:23

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