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第一讲 习题解答

习题1-1

1 计算下列极限

??1?p?① limn??1???1?,?p?0?

n??????n??p?1??1?1??11??1?0pp??????1?x?1?0????p?nn????lim?解：原式=lim??1?x??p ?limn??n??n?011x?0x?0?0nnpp??② limsinx?sina

x?asin?x?a?解：原式=limsinx?sina?x?a?sinx?sina?cosa =?sinx????limx?ax?ax?asinx?ax?a??????x?am③ limx?1nx?1，m,n为自然数 x?1m解：原式=limx?1x?1x?1??x?1nx?1?x??mx?1?1?x??nx?1?n m④ lim2?a?1,a?0

n??1??ln?2?an?1????lim?1n??n?n?n解：原式?limen??nln2a?1?n??e?ex?0limeln2ax?1x??

=ex?0limln2ax?1?ln2a0?1???x??e?ln?2a?1???xx?0?e2lna?a2

ax?xa,a?0 ⑤ limx?ax?aax?aa?aa?xaax?aaxa?aa??ax????xa???aa?lna?1? ?lim?lim解：原式=limx?ax?ax?ax?ax?ax?ax?ax?aaa?ax,a?0 ⑥ limxx?aa?xaxa解：原式?limxa?aa?ax?a?lim??limxax?aax?xax?ax?ax?aa?xaaaxaaxxaaxxaaa?aa?ax?aax?aaaxa?aa??x?a xaa?x?aa?aaax?aa?lim??x?a?x?ax?a??x?a?aa?aaax?aaxa?aa?x?a ??lim?aa?????ax?xax?a??x?a?ax?xax?ax?a????ax? ??a??ay???x?a???y?aa??x??a?1aa?a?lna ?a??x?a?a?lna?1??1?tgx?⑦ limx?010??1?sinx?

sinx1010?1?tgx?解：原式=limx?0??1?sinx?x ?xsinx10??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?????i?1n?mk??k??n?i?m????i?解：原式?lim????n???lim??n?1???n? m?1?n?????n??i?1??n?n?i?1??????i??1????1kkmk?k?1?m??n???i??1?x?? ?lim?i?

n??i2i?1i?1x?0nm??2 设f?x?在x0处二阶可导，计算limh?0f?x0?h??2f?x0??f?x0?h?。 2hf?x0?h??2f?x0??f?x0?h?f??x0?h??f??x0?h??lim解：lim

h?0h?0h22h?f??x0?h??f??x0?f??x0?h??f??x0??1? ?lim????f???x0? h?0h?h??23 设f??x0?存在，计算limx?x0xf?x0??x0f?x?

x?x0xf?x0??x0f?x0???x0f?x??x0f?x0????

解：原式?limx?x0x?x0 ?f?x0??limx?x0f?x??f?x0??x0?f?x0??x0f??x0?

x?x01lnx?lna?f?x??4 设a?0,f??a?存在，计算lim??x?afa????解：原式?ex?alimlnf?x??lnf?a?lnx?lna

af??a??ex?alimlnf?x??lnf?a?x?a?x?alnx?lna?ef?a?

习题1-2

1求下列极限 ① limsinx????x?1?sinx?1

?解：原式?limsinx?1?sinx?1??2?limsinx?2?0

x???????x?1?x?1????x???? ?介于x?1，与x?1之间。

cos?sinx??cosx② lim？？？？

x?0sin4x解：原式?lim?sin??sinx?x????0? 4x?0sinx?lim3??sinx?x??cosx?1sinx1?lim?lim?

x?0x?0x?06xx33x263ex?e?x③ lim

x?0tgx?sinx2ex?e?xex?e?xx??lim3?2?lim2e解：原式?lim???2 3x?0x?0x??x3??0x????xax?xa,a?0 ④ lim2x?ax?a2223333?ax?aa解：原式?lim?2x?a?x?a2?22??xa?aa???????x?a22?1? ??x?a?