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

From the given relation, I got that $f$ is infinitely differentiable. I tried to get an inequality between $x$ and $f(x)$, and so I tried to prove that the derivative doesn't change signs, so that $f$ is monotonic, but I don't think this works. Also, I noticed that if $f(x)=x$ for a given $x$, then $f'(x)=0$. I also don't see why the domain is given as $(1,\infty)$. Perhaps this is supposed to hint to something.

If $f(x)=0$ then $f'(x)=1$ and if $f'(x)=0$, then $f''(x)>0$. So $f$ has at most one local minimum, and either $f'$ is always negative (then so is $f$, and $f$ tends to $-\infty$ faster than $-x$) or $f'$ changes sign once from negative to positive or $f'$ is always positive. So $f$ is increasing on $(a,\infty)$ for some $a\ge 1$. Either $f$ tends to $\infty$ or $f$ is bounded. However, if $f$ is bounded, $f'(x)\to 1$ as $x\to\infty$, a contradiction.
