Schaum's Outline of Theory and Problems of Beginning Calculus

CHAP. 361 L'HOPITAL'S RULE; EXPONENTIAL GROWTH AND DECAY 29 136.20 Prove Cauchy's extended mean-value theorem: If f and g are differentiable in (a, b) and continuous on[a, b], and if g'(x) # 0 for all x in (a, b), then there exists a point c in (a, b) such that[Hint: Apply the generalized Rolle's theorem (Problem 17.19) toh(x) = (f(b) -f(a))g(x) - (g(b) - s(a))f(x)Note that g(b) # g(a) since, otherwise, the generalized Rolle's theorem would imply that g'(x) = 0 for some xin (a, b).)36.21 Prove L'H6pital's rule. [Hint: Consider the case lim,,,+ (f(x)/g(x)), where lim,,,, f(x) = lirn,,,, g(x) = 0.We may assume that f(a) = g(a) = 0. Then, by Problem 36.20, f(x)/g(x) = (f(x)-f(a))/@(x)- g(a)) =f'(x*)/g'(x*) for some x* between a and x. Therefore, as x +U+, x* -11'. Hence, if(f'(x)/g'(x)) = L, then lim,,,, (f(x)/g(x)) = L. The other cases can be handled in the same way orcan be reduced to this case.]