IMC 2001 · Problem 6

Suppose that the differentiable functions $a, b, f, g : \mathbb{R} \to \mathbb{R}$ satisfy

$$f(x) \ge 0, \; f'(x) \ge 0, \; g(x) > 0, \; g'(x) > 0 \text{ for all } x \in \mathbb{R},$$ $$\lim_{x \to \infty} a(x) = A > 0, \quad \lim_{x \to \infty} b(x) = B > 0, \quad \lim_{x \to \infty} f(x) = \lim_{x \to \infty} g(x) = \infty,$$

and

$$\frac{f'(x)}{g'(x)} + a(x) \frac{f(x)}{g(x)} = b(x).$$

Prove that

$$\lim_{x \to \infty} \frac{f(x)}{g(x)} = \frac{B}{A+1}.$$