Prove the Rational Limit Theorem.

First, before doing anything, note that $f(0,y) = 0$ whenever $y \neq 0,$ so if the limit exists, it must equal $0.$ With this context, we can see that (1) is the contrapositive of $\Rightarrow$ and (3) is $\Leftarrow.$ As you see in the hints below, (2) is simply a lemma to prove (3).

---

Now for the hints (really, proof sketches):

For 1, set $t = |x|^c = |y|^d$ and let $t \to 0^+$.

For 2, set $w = |x|^c, z = |y|^d, t = \frac{a}{c}$ in Young's theorem.

For 3, factor out $|y|^p$ for a particular $p>0$ (chosen so that we may apply part 2 to the quotient) to show $$f(x,y) = |y|^p \cdot \text{ Some bounded function}$$