I'm studying for an upcoming multivariable calculus exam and a question that has come up multiple times over the course of the semester has the following form. We consider $$f(x,y) = \cases{ \displaystyle{x^ay^b\over x^c + y^d},& if $(x,y)\neq (0,0)$\cr 0,& otherwise.}$$ and we're given $a,c,d$, so that only $b$ is unknown. First the question asks for a specific $b^*$ such that $f$ is continuous if $b>b^*$ and discontinuous if $b\leq b^*$. The second part of the question is the same, with "continuous" replaced by "differentiable". My questions are:
- Is there a general way of determining the correct values of $b^*$ just by eyeballing $a,c,d$ and avoiding nasty computations (which I often mess up)?
- Does this question work if we let $d$ be free and fix $a,b,c$ instead (in which case I imagine by symmetry of $x$ and $y$ it would also then work for $c$)? If so, is there a general formula in this case?
Thanks!
Extra details. I suppose I should say why I find this question annoying and why it'd be nice to have a general way to solve it. Both the "continuous" and "differentiable" boil down to showing that a limit goes to $0$. First we sort of have to guess the correct value of $b^*$ by splitting $f(x,y)$ into enough factors that setting $b$ to something makes the whole thing go to zero. Then to prove that it is discontinuous for all $b\leq b^*$, we have to come up with a clever algebraic curve on which to approach the origin such that the limit is not zero (and hope that this is actually possible and that we didn't mess up the first part). It just seems rather tricky and I'm sure there's a more robust approach, but I can't figure it out.