Let $f: \mathbb {R \to R}$ be a non-zero function such that $\displaystyle \lim_{x \to \infty} \frac{f(xy)}{x^3}$ exists for all $y>0$. Let $g(y) = \displaystyle \lim_{x \to \infty} \frac{f(xy)}{x^3}$ and $g(1)=1$, then what is $g(y)$ for all $y>0$?

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