Putnam 2009 B5

Let $f: (1, \infty) \to \mathbb{R}$ be a differentiable function such that $$f'(x) = \frac{x^2 - f(x)^2}{x^2 (f(x)^2 + 1)} \text{for all $x>1$.}$$ Prove that $\lim_{x \to \infty} f(x) = \infty$.