十次方了不起？洛（必达）他！

一、题目

$$
I=\lim \limits_{x \rightarrow 0} \frac{x^{2}-\int_{0}^{x^{2}} \cos t^{2} \mathrm{~d} t}{\sin ^{10} x}=
$$

难度评级：

二、解析

$$
I=\lim \limits_{x \rightarrow 0} \frac{x^{2}-\int_{0}^{x^{2}} \cos t^{2} d t}{\sin ^{10} x}=
$$

$$
\lim \limits_{x \rightarrow 0} \frac{x^{2}-\int_{0}^{x^{2}} \cos t^{2} d t}{x^{10}}=
$$

$$
\lim \limits_{x \rightarrow 0} \frac{2 x-2 x \cos x^{4}}{10 x^{9}}=
$$

$$
\lim \limits_{x \rightarrow 0} \frac{2 x\left(1-\cos x^{4}\right)}{10 x^{9}}=
$$

$$
\lim \limits_{x \rightarrow 0} \frac{2 x \cdot \frac{1}{2}\left(x^{4}\right)^{2}}{10 x^{9}}=
$$

$$
\lim \limits_{x \rightarrow 0} \frac{x^{9}}{10 x^{9}}=\frac{1}{10}.
$$

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