Question 107
The differentiable function $f : (0, \infty) \to \mathbb{R}$ is such that the limit $\lim_{x \to \infty} f'(x)$ exists and $x(f(x+1) - f(x)) = f(x)$, $\forall x \gt 0$. Prove that $f(x) = ax$, $\forall x \in (0, \infty)$, for some real number $a$.