Lim...it.

Consider a non-zero function $f: \mathbb {R \to R}$ such that $\displaystyle \lim_{x \to \infty} \frac {f(xy)}{x^3}$ exists for all $y>0$ . Let $\displaystyle g(y) = \lim_{x \to \infty} \frac {f(xy)}{x^3}$ . If $g(1)=1$ , find the value of $g(4)$ .