Problem 14 - Olympiad

First check continuity at 0: We need lim_(x→0) tan(x)/x = f(0) = 1. Using the standard limit lim_(x→0) tan(x)/x = 1 (since tan x ~ x as x → 0), the limit equals 1. Since f(0) = 1, we have lim_(x→0) f(x) = f(0), so f is continuous at 0. Now check differentiability: f'(0) = lim_(h→0) [f(h) - f(0)] / h = lim_(h→0) [tan(h)/h - 1] / h = lim_(h→0) [tan(h) - h] / h^2. Using the Taylor series: tan(h) = h + h^3/3 + ... So tan(h) - h = h^3/3 + ... Then [tan(h) - h] / h^2 = h/3 + ... → 0 as h → 0. Thus f'(0) = 0, and f is differentiable at 0. However, among the given choices, only choice A states that f is continuous at x = 0, which is true. Choices B, C, and D are incorrect. The correct answer is A.