Òò²»µÈʽÓë²»µÈʽµÈ¼Û£¬¹Êµ±n£¾Nʱ£¬ËùÓеĵãxn¶¼ÂäÔÚ¿ªÇø¼ä(a-¦Å£¬a+¦Å)ÄÚ£¬¶øÖ»ÓÐÓÐ
ÏÞ¸ö(ÖÁ¶àÖ»ÓÐN¸ö)ÔÚ´ËÇø¼äÒÔÍâ¡£
×¢£ºÓнçµÄÊýÁв»Ò»¶¨ÊÕÁ²£¬¼´£ºÊýÁÐÓнçÊÇÊýÁÐÊÕÁ²µÄ±ØÒªÌõ¼þ£¬µ«²»Êdzä·ÖÌõ¼þ¡£Àý£ºÊýÁÐ 1£¬-1£¬1£¬-1£¬¡£¬(-1)£¬¡ ÊÇÓнçµÄ£¬µ«ËüÊÇ·¢É¢µÄ¡£ 3¡¢ÊýÁм«Ï޵ļÆË㣨¿Î±¾Àý×Ó£©
n+1
¿Îºó×÷Òµ¼°Ð¡½á£º 1¡¢Ñ§Ï°ÁËÊýÁм«ÏÞ¸ÅÄî 2¡¢ÕÆÎÕÊýÁм«ÏÞÔËËã·½·¨¡£ ×÷Òµ£ºP15.2
µÚÈý½Ú£ºº¯Êý¼«Ï޵Ķ¨ÒåÓò¼ÆËã
Ç°ÃæÎÒÃÇѧϰÁËÊýÁеļ«ÏÞ£¬ÒѾ֪µÀÊýÁпɿ´×÷Ò»ÀàÌØÊâµÄº¯Êý£¬¼´×Ô±äÁ¿È¡ 1¡ú¡ÞÄÚµÄÕýÕûÊý£¬Èô×Ô±äÁ¿²»ÔÙÏÞÓÚÕýÕûÊýµÄ˳Ðò£¬¶øÊÇÁ¬Ðø±ä»¯µÄ£¬¾Í³ÉÁ˺¯Êý¡£ÏÂÃæÎÒÃÇÀ´Ñ§Ï°º¯ÊýµÄ¼«ÏÞ.
º¯ÊýµÄ¼«ÖµÓÐÁ½ÖÖÇé¿ö£ºa)£º×Ô±äÁ¿ÎÞÏÞÔö´ó£»b)£º×Ô±äÁ¿ÎÞÏÞ½Ó½üijһ¶¨µãx0£¬Èç¹ûÔÚÕâʱ£¬º¯ÊýÖµÎÞÏÞ½Ó½üÓÚijһ³£ÊýA£¬¾Í½Ð×öº¯Êý´æÔÚ¼«Öµ¡£ÎÒÃÇÒÑÖªµÀº¯ÊýµÄ¼«ÖµµÄÇé¿ö£¬ÄÇôº¯ÊýµÄ¼«ÏÞÈçºÎÄØ ?
ÏÂÃæÎÒÃǽáºÏ×ÅÊýÁеļ«ÏÞÀ´Ñ§Ï°Ò»Ïº¯Êý¼«Ï޵ĸÅÄî! 1¡¢º¯ÊýµÄ¼«ÏÞ(·ÖÁ½ÖÖÇé¿ö) a):×Ô±äÁ¿Ç÷ÏòÎÞÇî´óʱº¯ÊýµÄ¼«ÏÞ
¶¨Ò壺É躯Êýy=f(x)£¬Èô¶ÔÓÚÈÎÒâ¸ø¶¨µÄÕýÊý¦Å(²»ÂÛÆä¶àôС)£¬×Ü´æÔÚ×ÅÕýÊýX£¬Ê¹µÃ¶ÔÓÚÊʺϲ»µÈʽ Ëù¶ÔÓ¦µÄº¯ÊýÖµy=f(x)¶¼Âú×ã²»µÈʽ
µÄÒ»ÇÐx£¬
ÄÇÄ©³£ÊýA¾Í½Ð×öº¯Êýy=f(x)µ±x¡ú¡ÞʱµÄ¼«ÏÞ£¬¼Ç×÷£ºÏÂÃæÎÒÃÇÓñí¸ñ°Ñº¯ÊýµÄ¼«ÏÞÓëÊýÁеļ«ÏÞ¶Ô±Èһϣº ÊýÁеļ«Ï޵Ķ¨Òå º¯ÊýµÄ¼«Ï޵Ķ¨Òå ´æÔÚº¯Êýy=f(x)Óë³£ÊýA£¬ÈθøÒ»ÕýÊý¦Å£¾0£¬×Ü¿É´æÔÚÊýÁÐan=f(x)Óë³£ÊýA£¬ÈθøÒ»ÕýÊý¦Å£¾0£¬×Ü¿ÉÕÒµ½Ò»ÕýÕûÊýN£¬¶ÔÓÚn£¾NµÄËùÓÐan¶¼Âú×ãx¡ú¡ÞʱÊÕÁ²ÓÚA¼Ç£º¡£ £¼¦ÅÔò³ÆÊýÁÐan£¬µ±ÕÒµ½Ò»ÕýÊýX£¬¶ÔÓÚÊʺϵÄÒ»ÇÐx£¬¶¼Âú×㣬º¯Êýy=f(x)µ±x¡ú¡ÞʱµÄ¼«ÏÞΪA£¬¼Ç£º¡£ ´ÓÉϱíÎÒÃÇ·¢ÏÖÁËʲô £¿£¿ÊÔ˼¿¼Ö®
b):×Ô±äÁ¿Ç÷ÏòÓÐÏÞֵʱº¯ÊýµÄ¼«ÏÞ¡£ÎÒÃÇÏÈÀ´¿´Ò»¸öÀý×Ó. Àý£ºº¯Êý
£¬µ±x¡ú1ʱº¯ÊýÖµµÄ±ä»¯Ç÷ÊÆÈçºÎ£¿º¯ÊýÔÚx=1´¦ÎÞ¶¨Òå.ÎÒÃÇÖªµÀ¶ÔʵÊýÀ´½²£¬ÔÚÊýÖáÉÏÈκÎÒ»
¸öÓÐÏ޵ķ¶Î§ÄÚ£¬¶¼ÓÐÎÞÇî¶à¸öµã£¬Îª´ËÎÒÃÇ°Ñx¡ú1ʱº¯ÊýÖµµÄ±ä»¯Ç÷ÊÆÓñíÁгö,ÈçÏÂͼ:
×¢£ºÔÚ¶¨ÒåÖÐΪʲôÊÇÔÚÈ¥ÐÄÁÚÓòÄÚÄØ£¿ÕâÊÇÒòΪÎÒÃÇÖ»ÌÖÂÛx¡úx0µÄ¹ý³Ì£¬Óëx=x0³öµÄÇé¿öÎ޹ء£´Ë¶¨ÒåµÄºËÐÄÎÊÌâÊÇ£º¶Ô¸ø³öµÄ¦Å£¬ÊÇ·ñ´æÔÚÕýÊý¦Ä£¬Ê¹ÆäÔÚÈ¥ÐÄÁÚÓòÄÚµÄx¾ùÂú×ã²»µÈʽ¡£
ÓÐЩʱºò£¬ÎÒÃÇÒªÓô˼«Ï޵Ķ¨ÒåÀ´Ö¤Ã÷º¯ÊýµÄ¼«ÏÞΪ A£¬ÆäÖ¤Ã÷·½·¨ÊÇÔõÑùµÄÄØ£¿ a):ÏÈÈÎÈ¡¦Å£¾0£» b):д³ö²»µÈʽ
£¼¦Å£»
c):½â²»µÈʽÄÜ·ñµÃ³öÈ¥ÐÄÁÚÓò0£¼£¼¦Ä£¬ÈôÄÜ£»
d):Ôò¶ÔÓÚÈθøµÄ¦Å£¾0£¬×ÜÄÜÕÒ³ö¦Ä£¬µ±0£¼2¡¢º¯Êý¼«ÏÞµÄÔËËã¹æÔò
£¼¦Äʱ£¬£¼¦Å³ÉÁ¢£¬Òò´Ë
Ç°ÃæÒѾѧϰÁËÊýÁм«ÏÞµÄÔËËã¹æÔò£¬ÎÒÃÇÖªµÀÊýÁпÉ×÷ΪһÀàÌØÊâµÄº¯Êý£¬¹Êº¯Êý¼«ÏÞµÄÔËËã¹æÔòÓëÊýÁм«ÏÞµÄÔËËã¹æÔòÏàËÆ¡£
¢Å¡¢º¯Êý¼«ÏÞµÄÔËËã¹æÔò ÈôÒÑÖªx¡úx0(»òx¡ú¡Þ)ʱ£¬
.
Ôò£º
ÍÆÂÛ£º
ÔÚÇóº¯ÊýµÄ¼«ÏÞʱ£¬ÀûÓÃÉÏÊö¹æÔò¾Í¿É°ÑÒ»¸ö¸´Ôӵĺ¯Êý»¯ÎªÈô¸É¸ö¼òµ¥µÄº¯ÊýÀ´Çó¼«ÏÞ¡£
ÀýÌ⣺Çó
½â´ð£º
ÀýÌ⣺Çó
´ËÌâÈç¹ûÏñÉÏÌâÄÇÑùÇó½â£¬Ôò»á·¢Ïִ˺¯ÊýµÄ¼«ÏÞ²»´æÔÚ.ÎÒÃÇͨ¹ý¹Û²ì¿ÉÒÔ·¢ÏÖ´Ë·ÖʽµÄ·Ö×ӺͷÖĸ¶¼Ã»Óм«ÏÞ£¬ÏñÕâÖÖÇé¿öÔõô°ìÄØ£¿ÏÂÃæÎÒÃÇ°ÑËü½â³öÀ´¡£
½â´ð£º
×¢£ºÍ¨¹ý´ËÀýÌâÎÒÃÇ¿ÉÒÔ·¢ÏÖ£ºµ±·ÖʽµÄ·Ö×ӺͷÖĸ¶¼Ã»Óм«ÏÞʱ¾Í²»ÄÜÔËÓÃÉ̵ļ«ÏÞµÄÔËËã¹æÔòÁË£¬Ó¦ÏÈ°Ñ·ÖʽµÄ·Ö×Ó·Öĸת»¯Îª´æÔÚ¼«ÏÞµÄÇéÐΣ¬È»ºóÔËÓùæÔòÇóÖ®¡£
3¡¢×óÓÒ¼«ÏÞ¶¨Òå
¶¨Ò壺Èç¹ûx½ö´Ó×ó²à(x£¼x0)Ç÷½üx0ʱ£¬º¯Êýf(x)Óë³£Á¿AÎÞÏÞ½Ó½ü£¬Ôò³ÆAΪº¯Êýf(x)µ±
ʱµÄ×ó¼«ÏÞ.¼Ç£º
Èç¹ûx½ö´ÓÓÒ²à(x£¾x0)Ç÷½üx0ʱ£¬º¯Êýf(x)Óë³£Á¿AÎÞÏÞ½Ó½ü£¬Ôò³ÆAΪº¯Êýf(x)µ±×¢£ºÖ»Óе±x¡úx0ʱ£¬º¯Êýf(x)µÄ×ó¡¢ÓÒ¼«ÏÞ´æÔÚÇÒÏàµÈ£¬·½³Æf(x)ÔÚx¡úx0ʱÓм«ÏÞ ¿Îºó×÷Òµ¼°Ð¡½á£º
1¡¢Ñ§Ï°Á˺¯ÊýÊýÁм«ÏÞ¸ÅÄî 2¡¢ÕÆÎÕº¯ÊýÊýÁм«ÏÞÔËËã·½·¨¡£ ×÷Òµ£ºP23.1£¬2
ʱµÄÓÒ¼«ÏÞ.¼Ç£º
µÚËĽڣº¼«ÏÞÐÔÖÊ
1¡¢ÊýÁм«ÏÞµÄÐÔÖÊ
¶¨Àí1(¼«ÏÞµÄΨһÐÔ) ÊýÁÐ{xn}²»ÄÜÊÕÁ²ÓÚÁ½¸ö²»Í¬µÄ¼«ÏÞ? Ö¤Ã÷? ¼ÙÉèͬʱÓÐlimxn?a¼°limxn?b? ÇÒa
n??n??°´¼«Ï޵Ķ¨Òå? ¶ÔÓÚ??Òò´ËͬʱÓÐ xn?b?a>0? ´æÔÚ³ä·Ö´óµÄÕýÕûÊýN? ʹµ±n>Nʱ? ͬʱÓÐ|x?a|?b?a ¼°|x?b|?b?a?
nn
222b?a¼°x?b?a? n22ÕâÊDz»¿ÉÄܵÄ? ËùÒÔÖ»ÄÜÓÐa=b? ÊýÁеÄÓнçÐÔ? ¶ÔÓÚÊýÁÐ?xn}£¬Èç¹û´æÔÚ×ÅÕýÊýM£¬Ê¹µÃ¶ÔÒ»ÇÐxn¶¼Âú×ã ²»µÈʽ |xn|?M?Ôò³ÆÊýÁÐ{xn}ÊÇÓнçµÄ? Èç¹ûÕâÑùµÄÕýÊýM²»´æÔÚ£¬¾Í˵ÊýÁÐ{xn}ÊÇÎÞ½çµÄ
¶¨Àí2(ÊÕÁ²ÊýÁеÄÓнçÐÔ) Èç¹ûÊýÁÐ{xn}ÊÕÁ²? ÄÇôÊýÁÐ{xn}Ò»¶¨Óнç?
Ö¤Ã÷? ÉèÊýÁÐ{xn}ÊÕÁ²? ÇÒÊÕÁ²ÓÚa? ¸ù¾ÝÊýÁм«Ï޵Ķ¨Òå? ¶ÔÓÚ? ?1? ´æÔÚÕýÕûÊýN? ʹ¶ÔÓÚn>N ʱµÄÒ»ÇÐxn ? ²»µÈʽ |xn?a| ?1¶¼³ÉÁ¢? ÓÚÊǵ±n>Nʱ?
|xn|?|(xn ?a)?a| ?| xn?a|?|a|<1?|a|?
È¡M?max{|x 1|? |x 2|? ? ? ?? |x N |? 1?| a |}? ÄÇôÊýÁÐ{xn}ÖеÄÒ»ÇÐxn¶¼Âú×ã²»µÈʽ|xn|? M? Õâ¾ÍÖ¤Ã÷ÁËÊýÁÐ{xn}ÊÇÓнçµÄ?
¶¨Àí3£¨ÊÕÁ²ÊýÁеı£ºÅÐÔ) Èç¹ûÊýÁÐ{xn}ÊÕÁ²ÓÚa, ÇÒa?0(»òa?0)? ÄÇô´æÔÚÕýÕûÊýN? µ±n?Nʱ? ÓÐxn?0(»òxn?0)? Ö¤ ¾Ía?0µÄÇéÐÎÖ¤Ã÷? ÓÉÊýÁм«Ï޵Ķ¨Òå? ¶Ô??´Ó¶øxn?a?aa?0, ?N?N?, µ±n?Nʱ? ÓÐ|xn?a|?? 22aa??0? 22ÍÆÂÛ Èç¹ûÊýÁÐ{xn}´ÓijÏîÆðÓÐxn?0(»òxn?0)? ÇÒÊýÁÐ{xn}ÊÕÁ²ÓÚa? ÄÇôa?0(»òa?0).
Ö¤Ã÷ ¾Íxn?0ÇéÐÎÖ¤Ã÷? ÉèÊýÁÐ{xn}´ÓN1ÏîÆð? ¼´µ±n?N 1ʱÓÐxn?0? ÏÖÔÚÓ÷´Ö¤·¨Ö¤Ã÷? »òa?0? ÔòÓɶ¨Àí3Öª? ?N 2?N?, µ±n? N 2ʱ? ÓÐxn?0? È¡N?max{ N 1? N 2 }? µ±n?Nʱ? °´¼Ù¶¨ÓÐx n ?0? °´¶¨Àí3ÓÐx n?0? ÕâÒýÆðì¶Ü? ËùÒÔ±ØÓÐa ?0. ×ÓÊýÁÐ? ÔÚÊýÁÐ{xn}ÖÐÈÎÒâ³éÈ¡ÎÞÏÞ¶àÏî²¢±£³ÖÕâЩÏîÔÚÔÊýÁÐÖеÄÏȺó´ÎÐò? ÕâÑùµÃµ½µÄÒ»¸öÊýÁгÆΪÔÊýÁÐ{xn}µÄ×ÓÊýÁÐ? ÀýÈç? ÊýÁÐ{xn}? 1? ?1? 1? ?1? ? ? ?? (?1)n?1? ? ?µÄÒ»×ÓÊýÁÐΪ{x2n}? ?1? ?1? ?1? ? ? ?? (?1)2n?1? ? ? ¶¨Àí3(ÊÕÁ²ÊýÁÐÓëÆä×ÓÊýÁмäµÄ¹Øϵ) Èç¹ûÊýÁÐ{xn}ÊÕÁ²ÓÚa? ÄÇôËüµÄÈÎÒ»×ÓÊýÁÐÒ²ÊÕÁ²? ÇÒ¼«ÏÞÒ²ÊÇa ? Ö¤Ã÷? ÉèÊýÁÐ{xnk}ÊÇÊýÁÐ{xn}µÄÈÎÒ»×ÓÊýÁÐ?
ÒòΪÊýÁÐ{xn}ÊÕÁ²ÓÚa? ËùÒÔ?? >0? ?N?N+? µ±n?Nʱ? ÓÐ|xn?a|?? ?È¡K?N? Ôòµ±k?Kʱ? nk?k?K?N? ÓÚÊÇ|xnk?a|?? ? Õâ¾ÍÖ¤Ã÷ÁËlimxnk?a?
k??2¡¢º¯Êý¼«ÏÞµÄÐÔÖÊ
¶¨Àí1(º¯Êý¼«ÏÞµÄΨһÐÔ)Èç¹û¼«ÏÞlimf(x)´æÔÚ? ÄÇôÕ⼫ÏÞΨһ?
x?x0¶¨Àí2(º¯Êý¼«Ï޵ľֲ¿ÓнçÐÔ) Èç¹ûf(x)?A(x?x0)? ÄÇô´æÔÚ³£ÊýM?0ºÍ?? ʹµÃµ±0?|x?x0|??ʱ? ÓÐ|f(x)|?M? Ö¤Ã÷ ÒòΪf(x)?A(x?x0)? ËùÒÔ¶ÔÓÚ? ?1? ???0? µ±0?|x?x0|??ʱ? ÓÐ|f(x)?A|?? ?1?
ÓÚÊÇ |f(x)|?|f(x)?A?A|?|f(x)?A|?|A|?1?|A|?Õâ¾ÍÖ¤Ã÷ÁËÔÚx0µÄÈ¥ÐÄÁÚÓò{x| 0?|x?x0|?? }ÄÚ? f(x)ÊÇÓнçµÄ?
¶¨Àí3(º¯Êý¼«Ï޵ľֲ¿±£ºÅÐÔ) Èç¹ûf(x)?A(x?x0)? ¶øÇÒA?0(»òA?0)? ÄÇô´æÔÚ³£Êý??0? ʹµ±0?|x?x0|??ʱ? ÓÐf(x)?0(»òf(x)?0)?
¶¨Àí3? Èç¹ûf(x)?A(x?x0)(A?0)? ÄÇô´æÔÚµãx0µÄijһȥÐÄÁÚÓò? ÔÚ¸ÃÁÚÓòÄÚ? ÓÐ|f(x)|?|A|? ÍÆÂÛ Èç¹ûÔÚx0µÄijһȥÐÄÁÚÓòÄÚf(x)?0(»òf(x)?0)? ¶øÇÒf(x)?A(x?x0)? ÄÇôA?0(»òA?0)?
Ö¤Ã÷? Éèf(x)?0? ¼ÙÉèÉÏÊöÂ۶ϲ»³ÉÁ¢? ¼´ÉèA<0? ÄÇôÓɶ¨Àí1¾ÍÓÐx0µÄijһȥÐÄÁÚÓò? ÔÚ¸ÃÁÚÓòÄÚ f(x)?0? ÕâÓëf(x)?0µÄ¼Ù¶¨Ã¬¶Ü? ËùÒÔA?0?
¶¨Àí4(º¯Êý¼«ÏÞÓëÊýÁм«Ï޵ĹØϵ)
Èç¹ûµ±x?x0ʱf(x)µÄ¼«ÏÞ´æÔÚ? {xn}Ϊf(x)µÄ¶¨ÒåÓòÄÚÈÎÒ»ÊÕÁ²ÓÚx0µÄÊýÁÐ? ÇÒÂú×ãxn ?x0(n?N?)? ÄÇôÏàÓ¦µÄº¯ÊýÖµÊýÁÐ{f(x n)}±ØÊÕÁ²? ÇÒlimf(xn)?limf(x)?
n??x?x012Ö¤Ã÷ Éèf(x)?A(x?x0)? Ôò?? ?0? ?? ?0? µ±0?|x?x0|????ʱ? ÓÐ|f(x)?A|?? ? ÓÖÒòΪxn?x0(n??)? ¹Ê¶Ô? ?0? ?N?N?? µ±n?Nʱ? ÓÐ|xn?x0|?? ? ÓɼÙÉè? xn ?x0(n?N?)? ¹Êµ±n?Nʱ? 0?|x n?x 0|?? ? ´Ó¶ø|f(x n)?A|?? ? ¼´limf(xn)?limf(x)
n??x?x0¿Îºó×÷Òµ¼°Ð¡½á£º
1¡¢Ñ§Ï°Á˼«ÏÞµÄÏà¹Ø¶¨ÀíÓ뺯ÊýÁÐÏà¹Ø¶¨Àí ×÷Òµ£ºP30.8
µÚÎå½Ú£ºÁ½¸öÖØÒªµÄ¼«ÏÞ
1¡¢×¼ÔòI
Èç¹ûÊýÁÐ{xn }¡¢{yn}¼°{zn}Âú×ãÏÂÁÐÌõ¼þ? (1)yn?xn?zn(n?1? 2? 3? ? ? ?)? (2)limyn?a? limzn?a?
n??n??ÄÇôÊýÁÐ{xn }µÄ¼«ÏÞ´æÔÚ? ÇÒlimxn?a?
n??Ö¤Ã÷? ÒòΪlimyn?a? limzn?a? ÒÔ¸ù¾ÝÊýÁм«Ï޵Ķ¨Òå? ?? ?0? ?N 1?0? µ±n?N 1ʱ? ÓÐ
n??n??|y n?a|?? ? ÓÖ?N 2?0? µ±n?N 2ʱ? ÓÐ|z n?a|?? ? ÏÖÈ¡N?max{N 1? N 2}? Ôòµ± n?N ʱ? ÓÐ |y n?a|?? ? |z n?a|????ͬʱ³ÉÁ¢? ¼´
a???yn?a?? ? a???z n?a?? ?
ͬʱ³ÉÁ¢? ÓÖÒòyn?xn?zn ? ËùÒÔµ± n?N ʱ? ÓÐ a???yn?x n?z n?a?? ?
¼´ |x n?a|?? ? Õâ¾ÍÖ¤Ã÷ÁËlimxn?a?
n?? ¼òÒªÖ¤Ã÷? ÓÉÌõ¼þ(2)? ?? ?0? ?N ?0? µ±n?N ʱ? ÓÐ |y n?a|?? ¼°|z n?a|?? ? ¼´ÓÐ a???yn?a?? ? a???z n?a?? ? ÓÉÌõ¼þ(1)? ÓÐ
a???y n?x n?z n?a?? ?