Divide numerator and denominator by x: (x + sin x) / (x - cos x) = (1 + sin x / x) / (1 - cos x / x). As x → ∞, sin x / x → 0 and cos x / x → 0 (since |sin x| ≤ 1 and |cos x| ≤ 1). Therefore the limit is (1 + 0) / (1 - 0) = 1. The oscillatory behavior of sin x and cos x is dampened by division by x, which grows without bound. The correct answer is A.