当x→0时下列无穷小是x的n阶无穷小,求阶数n: (Ⅰ)一1; (Ⅱ)(1+tan2x)simx一1; (Ⅲ); (Ⅳ)∫0xsin.sin(1一cost)2dt.

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问题 当x→0时下列无穷小是x的n阶无穷小,求阶数n:
    (Ⅰ)一1;    (Ⅱ)(1+tan2x)simx一1;
    (Ⅲ);    (Ⅳ)∫0xsin.sin(1一cost)2dt.

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答案(Ⅰ)[*]一1~x4—2x2~一2x2 (x→0),即当x→0时[*]一1是x的2阶无穷小, 故n=2. (Ⅱ)(1+tan2x)sinx一1一ln[(1+tan2x)sinx一1+1] =sinxln(1+tan2x)~sinxtan2x~x.x2=x3 (x→0), 即当x→0时(1+tan2x)sinx一1是x的3阶无穷小,故n=3. (Ⅲ)由1一[*]是x的4阶无穷小,即当x→0时[*]是x的4阶无穷小,故n=4. [*] 即当x→0时∫0xsintsin(1一cost)2dt是x的6阶无穷小,故n=6.

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