ajoyib limitlar

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ajoyib limitlar ajoyib limitlar reja 1. funksiyaning limiti va uning asosiy xossalari. 2. aniqmasliklar va ularni ochish. 3. birinchi va ikkinchi ajoyib limitlar. 1. funksiyaning limiti va uning asosiy xossalari 1. 1-ta’rif. funksiya nuqtaning biror atrofida aniqlangan bo’lib, istalgan son uchun shunday son mavjud bo’lsaki, tengsizlikni qanoatlantiradigan barcha nuqtalar uchun tengsizlik bajarilsa, chekli son funksiyaning nuqtadagi limiti deb ataladi va quyidagicha yoziladi (1) funksiya limitining ta’rifidan kelib chiqadiki cheksiz kichik bo’lganda ham cheksiz kichik bo’ladi. 2-ta’rif. funksiya, ning yetarlicha katta qiymatlarida aniqlangan bo’lib, istalgan son uchun shunday, mavjud bo’lsaki, tengsizlikni qanoatlantiruvchi barcha lar uchun tengsizlik bajarilsa, o’zgarmas son, funksiyaning dagi limiti deyiladi, va (2) bilan belgilanadi. 1-ta’rifda faqat yoki bo’lgan qiymatlar qaralsa, funksiyaning chap yoki o’ng limit tushunchasi kelib chiqadi va , (3) bilan begilanadi. 3-ta’rif. limiti bo’lgan funksiyaga cheksiz kichik funksiya (ch. kich. f.) deyiladi. 4-ta’rif. limiti yoki bo’lgan funksiyalarga cheksiz katta funksiya (ch. kat. f.) deyiladi va (4) …

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ladi. 2. aniqmasliklar va ularni ochish 1.aniqmasliklar. limitni hisoblashda funksiyalar ch.kich.f. lar bo’lsa, nisbatga da (0/0) ko’rinishdagi aniqmaslik deyiladi. funksiyalar ch.kat.f. lar bo’lsa, nisbatga da ko’rinishidagi aniqmaslik deyiladi. xuddi shunga o’xshash aniqmasliklar limitlarni hisoblashda kelib chiqadi. bunday hollarda limitlarni hisoblashga aniqmasliklarni ochish deyiladi. va () ko’rinishdagi aniqmasliklarni ochishda quyidagi xossadan foydalaniladi: va funksiyalar nuqtaning biror atrofidagi hamma nuqtalarda o’zaro teng bo’lsa, ularning dagi limiti ham teng bo’ladi. masalan, va funksiyalar ning dan boshqa hamma qiymatlari uchun teng, chunki yuqoridagi xossaga asosan, bo’ladi, ya’ni natijaga ega bo’lamiz. funksiyalarning limitini topishga bir necha misollar qaraymiz. 1-misol. ekanligini funksiya limitining ta’rifidan foydalanib isbotlang. yyechish. buni isbotlash uchun o’zgaruvchi miqdor va o’zgarmas miqdor orasidagi farq da cheksiz kichik funksiya ekanligini ko’rsatish kifoya. demak, o’zgaruvchi miqdor da cheksiz kichik funksiyadan iborat. shunday qilib, . 2-misol. ekanligini isbotlang hamda va larning qiymatlari jadvali bilan tushuntiring. yechish. bo’lganligi uchun cheksiz kichik miqdordir. ni ayirmaga qo’yib, natijaga ega …

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ga asosan: hosil bњladi. yuqoridagi misolda, limitlarning xossalariga asosan, argument ning o’rniga uning limitik qiymatini qo’yishga olib keldi. 4-misol. limitni hisoblang. yechish. ikkita funksiya nisbatining limiti (8) formula hamda oldingi misolda foydalanilgan limitlarning xossalarini qo’llasak, bo’ladi. ratsional funksiyaing limitini hisoblash shu funksiyaning argument ning limitik qiymatidagi, qiymatini hisoblashga keltirildi. eslatma. elementar funksiyalarning intilgandagi limiti ( aniqlanish sohasiga tegishli) funksiyaning nuqtadagi qiymatiga teng bo’ladi. masalan, . 5-misol. limitni hisoblang. yechish. da surat ham, maxraj ham nolga aylanib ko’rinish-dagi aniqmaslik hosil bo’ladi. surat va maxrajni formula yordamida chiziqli ko’paytuvchilarga ajratamiz. bunda va lar kvadrat tenglamaning ildizlari. demak, bњladi. 6-misol. limitni hisoblang. yechish. da ko’rinishdagi aniqmas ifodaga ega bo’lamiz. bunday aniqmaslikni ochish uchun kasrning surat va maxrajini ning eng yuqori darajalisiga, ya’ni ga bo’lamiz, hamda limitlarning xossalaridan foydalansak bo’ladi. bunda lar da cheksiz kichik funksiyalardir. 7-misol. limitni hisoblang. yechish. da surat va maxraj 0 ga teng bo’ladi. maxrajda irratsional ifoda mavjud, uni suratga …

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tirishni bajaramiz: oxirgi ifoda da ko’rinishdagi aniqmaslik bo’lib, 11-misoldagidek ning yuqori darajalisiga surat va maxrajini bo’lib, bunda da bo’ladi. image2.wmf image47.wmf oleobject47.bin image48.wmf oleobject48.bin image49.wmf oleobject49.bin image50.wmf oleobject50.bin image51.wmf oleobject51.bin oleobject2.bin image52.wmf oleobject52.bin image53.wmf oleobject53.bin image54.wmf oleobject54.bin image55.wmf oleobject55.bin image56.wmf oleobject56.bin image3.wmf image57.wmf oleobject57.bin image58.wmf oleobject58.bin image59.wmf oleobject59.bin image60.wmf oleobject60.bin image61.wmf oleobject61.bin oleobject3.bin image62.wmf oleobject62.bin image63.wmf oleobject63.bin image64.wmf oleobject64.bin image65.wmf oleobject65.bin image66.wmf oleobject66.bin image4.wmf image67.wmf oleobject67.bin image68.wmf oleobject68.bin image69.wmf oleobject69.bin image70.wmf oleobject70.bin image71.wmf oleobject71.bin oleobject4.bin image72.wmf oleobject72.bin image73.wmf oleobject73.bin image74.wmf oleobject74.bin oleobject75.bin image75.wmf oleobject76.bin image76.wmf image5.wmf oleobject77.bin image77.wmf oleobject78.bin image78.wmf oleobject79.bin image79.wmf oleobject80.bin image80.wmf oleobject81.bin image81.wmf oleobject5.bin oleobject82.bin image82.wmf oleobject83.bin image83.wmf oleobject84.bin image84.wmf oleobject85.bin image85.wmf oleobject86.bin image86.wmf image6.wmf oleobject87.bin image87.wmf oleobject88.bin image88.wmf oleobject89.bin image89.wmf oleobject90.bin image90.wmf oleobject91.bin image91.wmf oleobject6.bin oleobject92.bin image92.wmf oleobject93.bin image93.wmf oleobject94.bin image94.wmf oleobject95.bin image95.wmf oleobject96.bin image96.wmf image7.wmf oleobject97.bin image97.wmf oleobject98.bin image98.wmf oleobject99.bin oleobject100.bin image99.wmf oleobject101.bin image100.wmf oleobject102.bin oleobject7.bin oleobject103.bin image101.wmf oleobject104.bin image102.wmf oleobject105.bin image103.wmf oleobject106.bin image104.wmf oleobject107.bin …

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e105.wmf image8.wmf oleobject108.bin image106.wmf oleobject109.bin image107.wmf oleobject110.bin oleobject111.bin image108.wmf oleobject112.bin image109.wmf oleobject113.bin oleobject8.bin image110.wmf oleobject114.bin image111.wmf oleobject115.bin image112.wmf oleobject116.bin image113.wmf oleobject117.bin image114.wmf oleobject118.bin image9.wmf image115.wmf image116.wmf image117.wmf oleobject119.bin image118.wmf oleobject120.bin image119.wmf oleobject121.bin image120.wmf oleobject122.bin oleobject9.bin image121.wmf oleobject123.bin image122.wmf oleobject124.bin image123.wmf oleobject125.bin image124.wmf oleobject126.bin image125.wmf oleobject127.bin image10.wmf image126.wmf oleobject128.bin image127.wmf oleobject129.bin image128.wmf oleobject130.bin image129.wmf oleobject131.bin image130.wmf oleobject132.bin oleobject10.bin image131.wmf oleobject133.bin image132.wmf oleobject134.bin image133.wmf oleobject135.bin image134.wmf oleobject136.bin image135.wmf oleobject137.bin image11.wmf image136.wmf oleobject138.bin image137.wmf oleobject139.bin image138.wmf oleobject140.bin image139.wmf oleobject141.bin image140.wmf oleobject142.bin oleobject11.bin image141.wmf oleobject143.bin image142.wmf oleobject144.bin image143.wmf oleobject145.bin image144.wmf oleobject146.bin image145.wmf oleobject147.bin image12.wmf image146.wmf oleobject148.bin image147.wmf oleobject149.bin image148.wmf oleobject150.bin image149.wmf oleobject151.bin image150.wmf oleobject152.bin oleobject12.bin image151.wmf oleobject153.bin image152.wmf oleobject154.bin image153.wmf oleobject155.bin image154.wmf oleobject156.bin image155.wmf oleobject157.bin image13.wmf image156.wmf oleobject158.bin image157.wmf oleobject159.bin image158.wmf oleobject160.bin image159.wmf oleobject161.bin oleobject162.bin image160.wmf oleobject13.bin oleobject163.bin image161.wmf oleobject164.bin image162.wmf oleobject165.bin image14.wmf oleobject14.bin image15.wmf oleobject15.bin image16.wmf image17.wmf oleobject16.bin imag

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ajoyib limitlar ajoyib limitlar reja 1. funksiyaning limiti va uning asosiy xossalari. 2. aniqmasliklar va ularni ochish. 3. birinchi va ikkinchi ajoyib limitlar. 1. funksiyaning limiti va uning asosiy xossalari 1. 1-ta’rif. funksiya nuqtaning biror atrofida aniqlangan bo’lib, istalgan son uchun shunday son mavjud bo’lsaki, tengsizlikni qanoatlantiradigan barcha nuqtalar uchun tengsizlik bajarilsa, chekli son funksiyaning nuqtadagi limiti deb ataladi va quyidagicha yoziladi (1) funksiya limitining ta’rifidan kelib chiqadiki cheksiz kichik bo’lganda ham cheksiz kichik bo’ladi. 2-ta’rif. funksiya, ning yetarlicha katta qiymatlarida aniqlangan bo’lib, istalgan son uchun shunday, mavjud bo’lsaki, tengsizlikni qanoatlantiruvchi barcha lar uchun tengsizlik bajarilsa, o’zgarmas son, funksiyaning ...

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