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:
- 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.
- Consider the behavior of $f$ along particular lines, such as $y = x$. I could not show that it was always above any $\varepsilon$.
- 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.
- 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.
- 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.
- 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 $2d < a + b$.
- 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.)