ISI 2016 B.Math Entrance (3)

Calculus Level 3

Let f:RRf: \mathbb {R \to R} be a non-zero function such that limxf(xy)x3\displaystyle \lim_{x \to \infty} \frac{f(xy)}{x^3} exists for all y>0y>0. Let g(y)=limxf(xy)x3g(y) = \displaystyle \lim_{x \to \infty} \frac{f(xy)}{x^3} and g(1)=1g(1)=1, then what is g(y)g(y) for all y>0y>0?

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