IMC 2001 Problem 6

Suppose that the differentiable functions $a, b, f, g:\mathbb{R} \rightarrow \mathbb{R}$ satisfy $$f(x)\geq 0, f'(x) \geq 0,g(x)\geq 0, g'(x) \geq 0 \text{ for all } x \in \mathbb{R},$$ $$\lim_{x\rightarrow \infty} a(x)=A\geq 0,\lim_{x\rightarrow \infty} b(x)=B\geq 0, \lim_{x\rightarrow \infty} f(x)=\lim_{x\rightarrow \infty} g(x)=\infty,$$ and $$\frac{f'(x)}{g'(x)}+a(x)\frac{f(x)}{g(x)}=b(x).$$ Prove that $\lim_{x\rightarrow\infty}\frac{f(x)}{g(x)}=\frac{B}{A+1}$ .