Let $\displaystyle f,g$ be two functions such that $\displaystyle {x^2}f\left( x \right) + {x^2}g\left( x \right) - {f^2}\left( x \right){g^2}\left( x \right) = 0$ for every $\displaystyle x > 0$. If $\displaystyle f,g $have positive values for every $\displaystyle x > 2012$, find the limits $\displaystyle \mathop {\lim }\limits_{x \to \infty } f\left( x \right),\mathop {\lim }\limits_{x \to \infty } g\left( x \right)$.

I firstly divided with $\displaystyle {x^2}{f^2}\left( x \right){g^2}\left( x \right)$, so that i could use this Squeeze theorem - Wikipedia, the free encyclopedia but with no result. Can you help?