V matematichnomu analizi verhnya i nizhnya granici viznachayutsya dlya chislovih poslidovnostej chi funkcij i vikoristovuyutsya pri yih vivchenni Na vidminu vid zvichajnoyi granici verhnya i nizhnya granici zavzhdi isnuyut hoch i mozhut buti rivnimi neskinchennosti Dlya nizhnoyi granici poslidovnosti x n n 1 displaystyle left x n right n 1 infty vikoristovuyutsya poznachennya lim n x n displaystyle varliminf n to infty x n poshirene v ukrayinskij i rosijskij literaturi i lim inf n x n displaystyle liminf n to infty x n poshirenishe v zahidnij literaturi Dlya verhnoyi granici vidpovidni poznachennya mayut viglyad lim n x n displaystyle varlimsup n to infty x n i lim sup n x n displaystyle limsup n to infty x n Verhnya granicya limsup i nizhnya granicya liminf Zmist 1 Viznachennya 1 1 Viznachennya dlya poslidovnostej 1 2 Viznachennya dlya funkcij 1 3 Viznachennya dlya poslidovnosti mnozhin 2 Prikladi 3 Vlastivosti 4 LiteraturaViznachennya RedaguvatiViznachennya dlya poslidovnostej Redaguvati Nizhnyu granicyu poslidovnosti mozhna viznachiti lim n x n lim n inf m n x m displaystyle varliminf n to infty x n lim n to infty Big inf m geq n x m Big nbsp abo lim n x n sup n 0 inf m n x m sup inf x m m n n 0 displaystyle varliminf n to infty x n sup n geq 0 inf m geq n x m sup inf x m m geq n n geq 0 nbsp Podibnim chinom verhnya granicya poslidovnosti xn viznachayetsya lim sup n x n lim n sup m n x m displaystyle limsup n to infty x n lim n to infty Big sup m geq n x m Big nbsp abo lim sup n x n inf n 0 sup m n x m inf sup x m m n n 0 displaystyle limsup n to infty x n inf n geq 0 sup m geq n x m inf sup x m m geq n n geq 0 nbsp Viznachennya dlya funkcij Redaguvati Nehaj dano dijsnu funkciyu f I R displaystyle f I to mathbb R nbsp de I R displaystyle I subset mathbb R nbsp i 3 granichnu tochku I todi verhnyu i nizhnyu granicyu funkciyi v tochci 3 mozhna viznachiti lim x 3 f x inf a gt 0 sup f 3 a 3 a I displaystyle varlimsup x to xi f x inf a gt 0 sup f xi a xi a cap I nbsp lim x 3 f x sup a gt 0 inf f 3 a 3 a I displaystyle varliminf x to xi f x sup a gt 0 inf f xi a xi a cap I nbsp Analogichno mozhna viznachiti odnostoronni granici funkciyi v tochci lim x 3 f x inf a gt 0 sup f 3 3 a I displaystyle varlimsup x to xi f x inf a gt 0 sup f xi xi a cap I nbsp lim x 3 f x sup a gt 0 inf f 3 3 a I displaystyle varliminf x to xi f x sup a gt 0 inf f xi xi a cap I nbsp lim x 3 f x inf a gt 0 sup f 3 a 3 I displaystyle varlimsup x to xi f x inf a gt 0 sup f xi a xi cap I nbsp lim x 3 f x sup a gt 0 inf f 3 a 3 I displaystyle varliminf x to xi f x sup a gt 0 inf f xi a xi cap I nbsp Viznachennya dlya poslidovnosti mnozhin Redaguvati Nehaj W deyaka mnozhina An poslidovnist yiyi pidmnozhin Todi verhnya i nizhnya granici ciyeyi poslidovnosti viznachayutsya za formulami lim n A n n 1 m n A m displaystyle varliminf n rightarrow infty A n bigcup n 1 infty left bigcap m n infty A m right nbsp i lim n A n n 1 m n A m displaystyle varlimsup n rightarrow infty A n bigcap n 1 infty left bigcup m n infty A m right nbsp Prikladi Redaguvatilim n 1 n lim n 1 n 0 displaystyle varliminf n to infty frac 1 n varlimsup n to infty frac 1 n 0 nbsp lim n 1 n 1 displaystyle varliminf n to infty left 1 right n 1 nbsp lim n 1 n 1 displaystyle varlimsup n to infty left 1 right n 1 nbsp Vlastivosti RedaguvatiU bud yakoyi poslidovnosti isnuyut verhnya i nizhnya granici sho nalezhat mnozhini R displaystyle mathbb R cup lbrace infty infty rbrace nbsp Chislova poslidovnist x n displaystyle x n nbsp zbigayetsya do a displaystyle a nbsp todi i tilki todi koli lim n x n lim n x n a displaystyle varliminf n rightarrow infty x n varlimsup n rightarrow infty x n a nbsp Dlya bud yakogo napered uzyatogo dodatnogo chisla e displaystyle varepsilon nbsp vsi elementi obmezhenoyi chislovoyi poslidovnosti x n n 1 displaystyle left x n right n 1 infty nbsp pochinayuchi z deyakogo nomera zalezhnogo vid e displaystyle varepsilon nbsp lezhat useredini intervalu lim n x n e lim n x n e displaystyle left varliminf n to infty x n varepsilon varlimsup n to infty x n varepsilon right nbsp Yaksho za mezhami intervalu a b displaystyle left a b right nbsp lezhit lishe skinchenna kilkist elementiv obmezhenoyi chislovoyi poslidovnosti x n n 1 displaystyle left x n right n 1 infty nbsp to interval lim n x n lim n x n displaystyle left varliminf n to infty x n varlimsup n to infty x n right nbsp mistitsya v intervali a b displaystyle left a b right nbsp Vikonuyutsya nerivnosti inf n x n lim inf n x n lim sup n x n sup n x n displaystyle inf n x n leq liminf n to infty x n leq limsup n to infty x n leq sup n x n nbsp lim n a n b n lim n a n lim n b n displaystyle varlimsup n to infty a n b n leq varlimsup n to infty a n varlimsup n to infty b n nbsp lim n a n lim n b n lim n a n b n displaystyle varliminf n to infty a n varliminf n to infty b n leq varliminf n to infty a n b n nbsp Literatura RedaguvatiGrigorij Mihajlovich Fihtengolc Kurs diferencialnogo ta integralnogo chislennya 2023 1100 s ukr Fihtengolc G M Kurs differencialnogo i integralnogo ischisleniya Moskva Nauka 1966 T 3 656 s ros Otrimano z https uk wikipedia org w index php title Verhnya i nizhnya granici amp oldid 38689208
