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? ?lim?ax??1?a2?x??1?limx?2?a??a2?x??2?12a?aa21??lna???
2??⑤ limn2?arctgn?????an?arctg?? n?1?aarctganan?arctg?an?1nan?1?an2解：原式?limn???n?n?1??lim?arctgx????0x???a?a
⑥ lim?1?x??1?1?3x?1?n?1??x?x?1?1?x?
x1?3x1?nx解：原式?lim ??x?11?x1?x1?xx?1x?1563?lim66x?1?6x?1x?15n?x?1x?1?limx?1??????x?3?x??nx??1n!
⑦ limx???x?x???6x?x 11??1?? xx?6解：原式?limx?61?x???1?1x?61?1x?2?lim??1 ?limx???1??1???1??x?x?1??6x?x???2?13
⑧ lim?nn?1?n?1nnln?n?1??2
n??1ln?n?1?lnn??1?ln?n?1?lnn?22nn?1e?en?????nnn?1?????lim解：原式?lim
n??n??ln?n?1?ln?n?1? ?limn?n?1?ln?n?1??nlnn2n???n?1?ln?n?1??limnln?n?1??nlnn?nln?n?1?22n???n?1?ln?n?1?
1??221nln?1???nln?n?1?n?nln?n?1?n??n?lim ?lim
n??n???n?1?ln?n?1??n?1?ln?n?1? ?limnn?1n???1?ln?n?1?ln?n?1?=1
n??1??fa?????n??? 2 设f?x?在a处可导，f?a??0，计算?lim?n??1?f?a??????n?????1??f?a??n??nln1??f?a??n??解：原式?limen???limen??1?lnf?a?n?1??a?n?1????lnf?a?n??1??????a??n??????.22f??a?=ef?a?
习题1-3
1 求下列极限
?1?x?① lim?x?0?1?x?解：原式?lim??1?1,n?0
??1?x?1???1?x?1?x?0???（也可利用对数等价关系求解）
② lim1?cosxcos2x???cosnx1?x?12
x?0解：利用等价代换可得 原式?lim?x?0cosxcos2x???cosnx?11?x?12?lim?x?0lncosxcos2x???cosnxln?1?x2?12
??2lim??lncosx?lncos2x?????lncosnxx2x?0
??2?lim?x?0??cosx?1x2?cos2x?1x2?????cosnx?1???? 2x???1?n2??1222?n?n?1??2n?1? ?1?2?????n ??2???2?????????6?2???2③ lim?x?0?1?x??? xe?1?1解：原式?lime?1?xx?e?1?xxx?0?lime?1?xx2xx?0?12