4.6.11. Let f and g be functions continuous and differentiable in (a, too), let limx- f (a) = limx-+ g(x) = oo, let g'(x) + 0 in (a, too), and let the limit limr- f'(x)/g'(x) exist (finite or infinite). Prove that the limit limx- f (x)/g(x) exists and that f (x) f' (ac ) lim lim roog(x) x-+00 g' (2)