2023年浙江大学强基计划数学试题（回忆版）#20
已知 $\left|\lim\limits_{x \rightarrow 0} \dfrac{\left.\ln \left(1+\sin ^2 x\right)-66 \sqrt[3]{2-\cos x}-1\right)}{x^4}\right|=\dfrac{q}{p}$，$p $ 和 $ q $ 是互素的正整数，则 $ p+q=$_____．
答案 19．
解析 根据题意，有qp=|limx→0ln(1+sin2x)−6(3√2−cosx−1)x4|=|limx→0sin2x−12sin4x+o(sin4x)−6(3√1+2sin2x2−1)x4|=|limx→0(x−16x3+o(x4))2−12x4+o(x4)−6(1+23sin2x2−49sin4x2+o(sin4x2)−1)x4|=|limx→0(x−16x3+o(x4))2−12x4+o(x4)−6(23sin2x2−49sin4x2+o(sin4x2))x4|=|limx→0x2−13x4+o(x4)−12x4+o(x4)−6(23(12x−148x3+o(116x4))2−136x4+o(x4))x4|=|limx→0x2−56x4+o(x4)−6(16x2−172x4+o(x4)−136x4+o(x4))x4|=|limx→0x2−56x4+o(x4)−6(16x2−124x4+o(x4))x4|=|limx→0x2−56x4+o(x4)−x2+14x4+o(x4)x4|=|limx→0−712x4+o(x4)x4|=|−712+limx→0o(x4)x4|=|−712|=712,于是 p+q=7+12=19．