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第一讲 习题解答

习题1-1

1 计算下列极限

p???1?① limn??1???1?,?p?0?

n??n??????解：原式=lim??1??n??1???1??1?n???limn??1np1?p?1?0pp????1?x???1?0?n?p??lim??1?x?n?01x?0?0np???p

x?0② limsinx?sinasin?x?a?

x?a解：原式=limsinx?sinax?a?x?a???x?a?sin?x?a??limsinx?sinax?a?x?a?=?sinx??x?a?cosa

m③ limx?1x?1x?1n，m,n为自然数

m解：原式=limx?1x?1x?1?x?1nx?1???mx?x?1?n1?x??x?1?nm

④ lim2?a?1n???n?n,a?0

1???ln2?an?1?????lim1n??解：原式?limen??nln2a?1?n?eln2a?1?e??ln?2a?1??xn?ex?0lim?x?

xx0ln2a?1?ln2a?1 =elimx?0???x??ex?0?e2lna?a

2⑤ limx?aa?xx?axa,a?0

xaaaxaaa解：原式=limx?aa?a?a?xx?axaa?lima?ax?ax?a?limx?ax?ax?a??ax??x?a??xa??x?a?aa?lna?1?

⑥ limx?aaax?aa?xx,a?0

解：原式?limxaax?axaax?aaa?xax?limaaax?axax?ax?a?x?aa?xxaaax?aaa?ax?ax?lima?xa?aaax?aa??x?aa?xxa

xa?aa?aaa?a?lim??x?a?x?a?x?axaaa?x?a?aa?aaa?ax?a?lim??a???xa?a?xax?a?x?ax?a??x?aa?x?a ??x?a?xa??ax? ??a??????ax?a10y??y?aa??xa??x?a?a1a?a?lna ??a?a?lna?1??⑦ lim?1?tgx???1?sinx?10x?0sinx

10解：原式=lim?1?tgx?10??1?sinx?xx?0?xsinx

?? ????1?tgx?10??1?tg0?10?1?sinx?10??1?sin0?10? ?lim?x?0?xx? ???1?tgx??10?x?0???1?sinx??10?x?0?20

?k?n?i?m??kn?，m为自然数 ⑧ lim??m?1n??n???i?1??k解：原式?lim??n????i?1k??n?i?m???n???lim???nm?1??n??i?1???mm???i??n?1???n? ???n???k ?limn???i?1i??1????1n??i??ink?i?1i??1?x??m??x?0?mk?k?1?2

2 设f?x?在x0处二阶可导，计算limf?x0?h??2f?x0??f?x0?h?h2。

h?0解：limf?x0?h??2f?x0??f?x0?h?h2h?0?limf??x0?h??f??x0?h?2hh?0

f??x0?h??f??x0??1?f??x0?h??f??x0?? ?lim????f???x0? h?0h?h??23 设f??x0?存在，计算limxf?x0??x0f?x?x?x0

x?x0xf?x0??x0f?x0????x0f?x??x0f?x0???解：原式?lim

x?x0x?x0 ?f?x0??limf?x??f?x0?x?x0x?x0?x0?f?x0??x0f??x0?

1?f?x??4 设a?0,f??a?存在，计算lim??x?afa????limx?alnx?lna

lnf?x??lnf?a?lnx?lna解：原式?e?ex?alimlnf?x??lnf?a?x?a?x?alnx?lnaaf??a??ef?a?

习题1-2

1求下列极限 ① limsinx????x?1?sinx?1

?解：原式?limsinx?1?sinx?1x????x?1???x?1??2?limsin?????x??x???2?0

?介于x?1，与x?1之间。

cos?sinx??cosxsinx4② lim？？？？

x?0解：原式?lim?sin??sinx?x?sinx??sinx?x?x?x3x?04???0?

?lim3x?03?lim?cosx?13x2x?0?limsinx6xx?0?16

③ limex?ex?0tgx?sinx2

?x3解：原式?limex3?ex3x?0?limex3?e?x3x?0x???x33??2?lim2?e??0x????x?2

④ limax2?xa22x?ax?a2,a?0

22?ax?aa解：原式?lim?22x?a?x?a???xa?aa?????x?a??22?1 ???x?a?