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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

$$\sup_{x\in \mathbb R} \frac{f(x,y)}{g(x,y)} \leq \sup_{x\in \mathbb R} f(x,y) . \sup_{x\in \mathbb R} 1/g(x,y)$$ $$\leq \sup_{x\in \mathbb R} f(x,y) .\frac{1}{\inf_{x\in \mathbb R} g(x,y)}$$


$$\limsup_{y\longrightarrow \infty}\; \sup_{x\in \mathbb R} \frac{f(x,y)}{g(x,y)}\leq \limsup_{y\longrightarrow \infty}\;\sup_{x\in \mathbb R} f(x,y) .\limsup_{y\longrightarrow \infty}\;\frac{1}{\inf_{x\in \mathbb R} g(x,y)}$$

and here I got stuck!

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What have you tried so far? It's always easier for people to help to see what you have done so far, also this approach is helps you see why you got stuck or where you went wrong. – sxd Dec 26 '11 at 2:20
Word of caution: $\sup(1/g(x,y))=1/\inf g(x,y)$ holds as long as $g$ is either always positive or always negative. And of course never approaches $0$ from above or $- \infty$ – Arthur Dec 26 '11 at 3:30
So no help!!!... – Monica Dec 27 '11 at 15:30

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