Odnostoronnya granicya v matematichnomu analizi granicya funkciyi dijsnoyi zminnoyi yaka peredbachaye pryamuvannya do granichnoyi tochki tilki z odnogo boku zliva abo sprava Taki granici nazivayut vidpovidno livostoronnoyu graniceyu abo livoyu graniceyu ta pravostoronnoyu graniceyu abo pravoyu graniceyu Funkciya f x x 2 sign x displaystyle f x x 2 operatorname sign x maye livostoronnyu granicyu 1 displaystyle 1 pravostoronnyu granicyu 1 displaystyle 1 i znachennya funkciyi rivne 0 displaystyle 0 u tochci x 0 displaystyle x 0 Zmist 1 Oznachennya 1 1 Oznachennya za Koshi 1 2 Oznachennya za Gejne 2 Prikladi 3 Div takozh 4 LiteraturaOznachennya RedaguvatiIsnuye kilka rivnosilnih viznachen granici funkciyi v tochci sered nih ye sformulovani Koshi ta Gejne Nehaj A R displaystyle A subset mathbb R nbsp prichomu A displaystyle A neq emptyset nbsp i x 0 displaystyle x 0 nbsp granichna tochka mnozhini A displaystyle A nbsp U podalshomu budemo rozglyadati funkciyi f A R displaystyle f A to mathbb R nbsp Oznachennya za Koshi Redaguvati Oznachennya pravostoronnoyi granici Nehaj x 0 displaystyle x 0 nbsp taka granichna tochka mnozhini A displaystyle A nbsp sho isnuye g gt 0 displaystyle gamma gt 0 nbsp take sho x 0 x 0 g A displaystyle x 0 x 0 gamma subset A nbsp Chislo a displaystyle a nbsp nazivayetsya pravostoronnoyu graniceyu funkciyi f displaystyle f nbsp v tochci x 0 displaystyle x 0 nbsp yaksho dlya dovilnogo dodatnogo e displaystyle varepsilon nbsp isnuye dodatne chislo d displaystyle delta nbsp take sho dlya dovilnogo x x 0 x 0 d displaystyle x in x 0 x 0 delta nbsp vikonuyetsya f x a lt e displaystyle f x a lt varepsilon nbsp Pravostoronnyu granicyu prijnyato poznachati nastupnim chinom lim x x 0 f x lim x x 0 0 f x lim x x 0 f x lim x x 0 f x displaystyle lim limits x to x 0 f x lim limits x to x 0 0 f x lim x downarrow x 0 f x lim x searrow x 0 f x nbsp Oznachennya livostoronnoyi granici Nehaj x 0 displaystyle x 0 nbsp taka granichna tochka mnozhini A displaystyle A nbsp sho isnuye g gt 0 displaystyle gamma gt 0 nbsp take sho x 0 g x 0 A displaystyle x 0 gamma x 0 subset A nbsp Chislo a displaystyle a nbsp nazivayetsya livostoronnoyu graniceyu funkciyi f displaystyle f nbsp v tochci x 0 displaystyle x 0 nbsp yaksho dlya dovilnogo dodatnogo e displaystyle varepsilon nbsp isnuye dodatne chislo d displaystyle delta nbsp take sho dlya dovilnogo x x 0 d x 0 displaystyle x in x 0 delta x 0 nbsp vikonuyetsya f x a lt e displaystyle f x a lt varepsilon nbsp Dlya livostoronnoyi granici prijnyati taki poznachennya lim x x 0 f x lim x x 0 0 f x lim x x 0 f x lim x x 0 f x displaystyle lim limits x to x 0 f x lim limits x to x 0 0 f x lim x uparrow x 0 f x lim x nearrow x 0 f x nbsp Vikoristovuyutsya takozh nastupni skorochennya f x 0 displaystyle f left x 0 right nbsp i f x 0 0 displaystyle f left x 0 0 right nbsp dlya pravoyi granici f x 0 displaystyle f left x 0 right nbsp i f x 0 0 displaystyle f left x 0 0 right nbsp dlya livoyi granici Oznachennya za Gejne Redaguvati Oznachennya pravostoronnoyi granici Nehaj x 0 displaystyle x 0 nbsp taka granichna tochka mnozhini A displaystyle A nbsp sho isnuye g gt 0 displaystyle gamma gt 0 nbsp take sho x 0 x 0 g A displaystyle x 0 x 0 gamma subset A nbsp Chislo p displaystyle p nbsp nazivayetsya pravostoronnoyu graniceyu funciyi f displaystyle f nbsp v tochci x 0 displaystyle x 0 nbsp yaksho dlya bud yakoyi poslidovnosti a n n 0 A displaystyle a n n 0 infty subset A nbsp a n gt x 0 displaystyle a n gt x 0 nbsp pri n N displaystyle n in mathbb N nbsp sho zbigayetsya do chisla x 0 displaystyle x 0 nbsp vidpovidna poslidovnist znachen funkciyi f x n n 1 displaystyle left f left x n right right n 1 infty nbsp zbizhna i maye graniceyu odne i tezh same chislo p displaystyle p nbsp Oznachennya livostoronnoyi granici Nehaj x 0 displaystyle x 0 nbsp taka granichna tochka mnozhini A displaystyle A nbsp sho isnuye g gt 0 displaystyle gamma gt 0 nbsp take sho x 0 g x 0 A displaystyle x 0 gamma x 0 subset A nbsp Chislo p displaystyle p nbsp nazivayetsya pravostoronnoyu graniceyu funciyi f displaystyle f nbsp v tochci x 0 displaystyle x 0 nbsp yaksho dlya bud yakoyi poslidovnosti a n n 0 A displaystyle a n n 0 infty subset A nbsp a n lt x 0 displaystyle a n lt x 0 nbsp pri n N displaystyle n in mathbb N nbsp sho zbigayetsya do chisla x 0 displaystyle x 0 nbsp vidpovidna poslidovnist znachen funkciyi f x n n 1 displaystyle left f left x n right right n 1 infty nbsp zbizhna i maye graniceyu odne i tezh same chislo p displaystyle p nbsp Yaksho obidvi odnostoronni granici isnuyut v tochci x 0 displaystyle x 0 nbsp ta rivni v nij to mozhna pokazati sho lim x x 0 f x lim x x 0 f x lim x x 0 f x displaystyle lim limits x to x 0 f x lim limits x to x 0 f x lim limits x to x 0 f x nbsp Yaksho odnostoronni granici isnuyut v tochci x 0 displaystyle x 0 nbsp ale ne rivni to granici v tochci x 0 displaystyle x 0 nbsp ne isnuye Yaksho bud yaka odnostoronnya granicya ne isnuye to i granici takozh ne isnuye Prikladi RedaguvatiPriklad 1 Livoyu ta pravoyu granicyami funkciyi g x 1 x displaystyle g x frac 1 x nbsp pri x 0 displaystyle x to 0 nbsp ye lim x 0 1 x displaystyle lim x to 0 1 x infty nbsp ta lim x 0 1 x displaystyle lim x to 0 1 x infty nbsp Prichina chomu lim x 0 1 x displaystyle lim x to 0 1 x infty nbsp v tomu sho x displaystyle x nbsp vid yemnij pri x 0 displaystyle x to 0 nbsp sho v comu vipadku oznachaye sho 1 x displaystyle 1 x nbsp dodatnya tomu lim x 0 1 x displaystyle lim x to 0 1 x nbsp rozhoditsya do displaystyle infty nbsp Analogichno lim x 0 1 x displaystyle lim x to 0 1 x infty nbsp bo x displaystyle x nbsp dodatnij pri x 0 displaystyle x to 0 nbsp sho v comu vipadku oznachaye sho 1 x displaystyle 1 x nbsp vid yemna tomu lim x 0 1 x displaystyle lim x to 0 1 x nbsp rozhoditsya do displaystyle infty nbsp nbsp Grafik funkciyi 1 1 2 1 x displaystyle 1 1 2 1 x nbsp Priklad 2 Odnim iz prikladiv funkcij z riznimi odnostoronnimi granicyami ye f x 1 1 2 1 x displaystyle f x frac 1 1 2 1 x nbsp dlya yakoyi liva granicya dorivnyuye lim x 0 f x 0 displaystyle lim x to 0 f x 0 nbsp a prava granicya lim x 0 f x 1 displaystyle lim x to 0 f x 1 nbsp Vikoristovuyuchi poperednij priklad otrimuyemo lim x 0 2 1 x displaystyle lim x to 0 2 1 x infty nbsp ta lim x 0 2 1 x 0 displaystyle lim x to 0 2 1 x 0 nbsp Tomu lim x 0 1 1 2 1 x 1 1 lim x 0 2 1 x 1 1 0 1 displaystyle lim x to 0 frac 1 1 2 1 x frac 1 1 displaystyle lim x to 0 2 1 x frac 1 1 0 1 nbsp a lim x 0 1 1 2 1 x 0 displaystyle lim x to 0 frac 1 1 2 1 x 0 nbsp bo znamennik pryamuye do neskinchennosti tobto lim x 0 1 2 1 x displaystyle lim x to 0 1 2 1 x infty nbsp Otzhe lim x 0 f x lim x 0 f x displaystyle lim x to 0 f x neq lim x to 0 f x nbsp a granici lim x 0 f x displaystyle lim x to 0 f x nbsp ne isnuye Div takozh RedaguvatiGranicya funkciyi v tochciLiteratura RedaguvatiGrigorij Mihajlovich Fihtengolc Kurs diferencialnogo ta integralnogo chislennya 2023 1300 s ukr Dorogovcev A Ya Matematichnij analiz Pidruchnik U dvoh chastinah Chastina 1 Kiyiv Libid 1993 320 s ISBN 5 325 00380 1 Otrimano z https uk wikipedia org w index php title Odnostoronnya granicya amp oldid 38689148
