Given $f(x,y),g(x,y)$, positive functions of $x,y\in \mathbb R$. can we write $$\limsup_{y\longrightarrow \infty}\; \sup_{x\in \mathbb R} \frac{f(x,y)}{g(x,y)}$$
in terms of a product of $f(x,y), g(x,y)$ instead of $\frac{f(x,y)}{g(x,y)}$?
(Note: You can assume all limits exist and nonzero.)
Progress: I believe that $\sup(1/g(x,y))=1/\inf g(x,y)$, so
$$\sup_{x\in \mathbb R} \frac{f(x,y)}{g(x,y)} \leq \sup_{x\in \mathbb R} f(x,y) . \sup_{x\in \mathbb R} 1/g(x,y)$$ $$\leq \sup_{x\in \mathbb R} f(x,y) .\frac{1}{\inf_{x\in \mathbb R} g(x,y)}$$
so:
$$\limsup_{y\longrightarrow \infty}\; \sup_{x\in \mathbb R} \frac{f(x,y)}{g(x,y)}\leq \limsup_{y\longrightarrow \infty}\;\sup_{x\in \mathbb R} f(x,y) .\limsup_{y\longrightarrow \infty}\;\frac{1}{\inf_{x\in \mathbb R} g(x,y)}$$
and here I got stuck!