第十四次课 20环境工程1环境科学1、2班113（124）人 ① 罗比达法则
② 非基础形式的未定式的处理 ③ 函数的单调性
limf(x)?0,limF(x)?0,?x?ax?a?f(x)f?(x)?f(x),F?x??diff?x??a???lim?limx?aF(x)x?aF?(x)?F??x??0?????????????????????????sup?f(a)?F?a??0.f(x),F?x??on?a,x??or??x,a???c.t???f(x),F?x??in?a,x??or??x,a???diff.??????a,x??or??x,a??,??F?x??0??f????f?x??f?a?f(x)f????????F????F?x??F?a?F(x)F????limx?af????f????f??x?f(x)?lim?lim?limF(x)x?aF??????aF????x?aF??x??2
?arctanx??????.1x?????x1?2x21?x?lim?lim x???x???1?x2?1x22x?lim?1x???2xlimlnx(n?0).x???xn
1?lim?0x???nxnlimxn???lim?x(n?N? ,??0).??x???e????limn?n?1?xnx?limx????e?xx????2e?xn?1n?2???limn!
x????en?x?0对数、幂函数、指数三类无穷大量，速度最快的是指数、最慢是对数，
limf(x)??,limF(x)??,?x?ax?a?f(x)f?(x)?f(x),F?x??diff??lim?lim?x?aF(x)x?aF?(x)?F??x??0????????????????????????????F??x?1?2?f2(x)F??x??f(x)???F(x)???F(x)lim?lim?2? ???lim????limx?aF(x)?x?ax?ax?a1??fxF(x)fx???????????2f(x)f(x)?limx?af(x)f(x)F??x?f(x)F??x????lim??1x?aF(x)F(x)f??x?F(x)f??x?f??x?1?limF??x?x?aF??x?limx?af??x?f(x)?lim?x?aF(x)--------------------------------+----------- 基础未定式
非基础型未定式
????③?????④?-???⑤??0???⑥?00???⑦?1??????③????①??or?②???????110?0?④???????21①01020102?①???②⑤??0???⑥?00???⑦?1??对数恒等式?
??????⑤??0?对数恒等式?exp?0?ln???③????①??or?②???????????⑥?00???对数恒等式?exp?0?ln0??③????①??or?②???????????⑦?1??对数恒等式?exp???ln1??③????①??or?②???????limx2e1/xx?0121e2ex?lim?limx?01x?0?2x??
x2?1??1?x2?
???limex2??x?01??x2??????limx?1ln1?x????????1?x?t,x?1,t?0???????x?1lntt2??limtlnt??lim?lim?limt?0t?0?t?0?1t?0?tt?0?t??lim?x?1?ln?1?x???x?1
12ln?1?x?x?1??1?x?lim?lim?lim?011x?1?x?1?x?1?1?x2?x?1??x?1?
1??1lim??x?x?0?xe?1?ex?1?x????lim?x?0xex?1??????x21?x????x2??1?xxe?1?x12!?lim?lim?x?0x?0x2x22 x2e?1?x????x2?2!xex?1ex1?lim?lim?x?02xx?0221?2
1?sinx?x2lim?? x?0?x?
1??2lim?2??x?1?x?1x?1?2?x?1?0??lim2???? x?1x?1?0??11?lim??x?12x2
2021年高等数学课件-第十四次课 20环境工程1环境科学1
