Ce spisok granic dlya poshirenih funkcij Zmist 1 Polinomi ta funkciyi viglyadu xa 1 1 Polinomi 1 2 Funkciyi viglyadu xa 2 Eksponencialni funkciyi 2 1 Funkciyi viglyadu ag x 2 2 Funkciyi viglyadu xg x 2 3 Funkciyi viglyadu f x g x 2 4 Chastki dobutki ta kompoziti 3 Logarifmichni funkciyi 3 1 Naturalnij logarifm 3 2 Logarifmi z dovilnoyu osnovoyu 4 Trigonometrichni funkciyi 5 Dzherela 6 PrimitkiPolinomi ta funkciyi viglyadu xa Redaguvatilim x c a a displaystyle lim x to c a a nbsp dlya vsih c a R displaystyle c a in mathbb R nbsp Polinomi Redaguvati lim x c x c displaystyle lim x to c x c nbsp dlya vsih c R displaystyle c in mathbb R nbsp lim x c a x b a c b displaystyle lim x to c ax b ac b nbsp dlya vsih a b c R displaystyle a b c in mathbb R nbsp lim x c x n c n displaystyle lim x to c x n c n nbsp dlya vsih n N displaystyle n in mathbb N nbsp c R displaystyle c in mathbb R nbsp lim x x a a gt 0 ne isnuye a 0 a lt 0 displaystyle lim x to infty x a begin cases infty amp a gt 0 text ne isnuye amp a 0 infty amp a lt 0 end cases nbsp dlya vsih a R displaystyle a in mathbb R nbsp Zagalom yaksho p x displaystyle p x nbsp ye polinomom to za neperervnistyu polinomiv lim x c p x p c displaystyle lim x to c p x p c nbsp Ce takozh spravedlivo dlya racionalnih funkcij oskilki voni neperervni u svoyij oblasti viznachennya Funkciyi viglyadu xa Redaguvati lim x c x a c a displaystyle lim x to c x a c a nbsp Zokrema lim x x a a gt 0 1 a 0 0 a lt 0 displaystyle lim x to infty x a begin cases infty amp a gt 0 1 amp a 0 0 amp a lt 0 end cases nbsp lim x c x 1 a c 1 a displaystyle lim x to c x 1 a c 1 a nbsp Zokrema lim x x 1 a lim x x a displaystyle lim x to infty x 1 a lim x to infty sqrt a x infty nbsp dlya bud yakogo a gt 0 displaystyle a gt 0 nbsp 1 lim x 0 x n lim 1 x n displaystyle lim x to 0 x n lim frac 1 x n infty nbsp lim x 0 x n lim x 0 1 x n n neparne n parne displaystyle lim x to 0 x n lim x to 0 frac 1 x n begin cases infty amp n text neparne infty amp n text parne end cases nbsp lim x a x 1 lim x a x 0 displaystyle lim x to infty ax 1 lim x to infty a x 0 nbsp dlya dovilnogo a R displaystyle a in mathbb R nbsp Eksponencialni funkciyi RedaguvatiFunkciyi viglyadu ag x Redaguvati lim x c e x e c displaystyle lim x to c e x e c nbsp lim x a x a gt 1 1 a 1 0 0 lt a lt 1 displaystyle lim x to infty a x begin cases infty amp a gt 1 1 amp a 1 0 amp 0 lt a lt 1 end cases nbsp lim x a x 0 a gt 1 1 a 1 0 lt a lt 1 displaystyle lim x to infty a x begin cases 0 amp a gt 1 1 amp a 1 infty amp 0 lt a lt 1 end cases nbsp lim x a x lim x a 1 x 1 a gt 0 0 a 0 ne isnuye a lt 0 displaystyle lim x to infty sqrt x a lim x to infty a 1 x begin cases 1 amp a gt 0 0 amp a 0 text ne isnuye amp a lt 0 end cases nbsp Funkciyi viglyadu xg x Redaguvati lim x x x lim x x 1 x 1 displaystyle lim x to infty sqrt x x lim x to infty x 1 x 1 nbsp Funkciyi viglyadu f x g x Redaguvati lim x 1 1 x x lim x 0 1 x 1 x e displaystyle lim x to infty left 1 frac 1 x right x lim x to 0 left 1 x right frac 1 x e nbsp druga chudova granicya 2 lim x 1 1 x x 1 e displaystyle lim x to infty left 1 frac 1 x right x frac 1 e nbsp lim x 1 k x m x lim x 0 1 k x m x e m k displaystyle lim x to infty left 1 frac k x right mx lim x to 0 left 1 kx right frac m x e mk nbsp lim x x x k x e k displaystyle lim x to infty left frac x x k right x e k nbsp lim x 0 1 a e x 1 1 x e a displaystyle lim x to 0 left 1 a left e x 1 right right frac 1 x e a nbsp Chastki dobutki ta kompoziti Redaguvati lim x 0 x e x 0 displaystyle lim x to 0 xe x 0 nbsp lim x x e x 0 displaystyle lim x to infty xe x 0 nbsp lim x 0 a x 1 x ln a displaystyle lim x to 0 left frac a x 1 x right ln a nbsp dlya vsih a gt 0 displaystyle a gt 0 nbsp lim x 0 e x 1 x 1 displaystyle lim x to 0 left frac e x 1 x right 1 nbsp lim x 0 e a x 1 x a displaystyle lim x to 0 left frac e ax 1 x right a nbsp dlya vsih a gt 0 displaystyle a gt 0 nbsp Logarifmichni funkciyi RedaguvatiNaturalnij logarifm Redaguvati lim x c ln x ln c displaystyle lim x to c ln x ln c nbsp dlya vsih c gt 0 displaystyle c gt 0 nbsp Zokrema lim x 0 log x displaystyle lim x to 0 log x infty nbsp lim x log x displaystyle lim x to infty log x infty nbsp lim x 1 ln x x 1 1 displaystyle lim x to 1 frac ln x x 1 1 nbsp lim x 0 ln x 1 x 1 displaystyle lim x to 0 frac ln x 1 x 1 nbsp lim x 0 ln 1 a e x 1 x a displaystyle lim x to 0 frac ln left 1 a left e x 1 right right x a nbsp vivoditsya za pravilom Lopitalya lim x 0 x ln x 0 displaystyle lim x to 0 x ln x 0 nbsp lim x ln x x 0 displaystyle lim x to infty frac ln x x 0 nbsp Logarifmi z dovilnoyu osnovoyu Redaguvati Dlya b gt 1 lim x 0 log b x displaystyle lim x to 0 log b x infty nbsp lim x log b x displaystyle lim x to infty log b x infty nbsp Dlya b lt 1 lim x 0 log b x displaystyle lim x to 0 log b x infty nbsp lim x log b x displaystyle lim x to infty log b x infty nbsp Dlya oboh vipadkiv mozhna uzagalniti lim x 0 log b x F b displaystyle lim x to 0 log b x F b infty nbsp lim x log b x F b displaystyle lim x to infty log b x F b infty nbsp de F x 2 H x 1 1 displaystyle F x 2H x 1 1 nbsp i H x displaystyle H x nbsp funkciya Gevisajda Trigonometrichni funkciyi Redaguvatilim x a sin x sin a displaystyle lim x to a sin x sin a nbsp lim x a cos x cos a displaystyle lim x to a cos x cos a nbsp lim x 0 sin x x 1 displaystyle lim x to 0 frac sin x x 1 nbsp persha chudova granicya Uzagalnennya lim x 0 sin a x a x 1 displaystyle lim x to 0 frac sin ax ax 1 nbsp pri a 0 displaystyle a neq 0 nbsp lim x 0 sin a x x a displaystyle lim x to 0 frac sin ax x a nbsp dlya vsih a R displaystyle a in mathbb R nbsp lim x 0 sin a x b x a b displaystyle lim x to 0 frac sin ax bx frac a b nbsp pri b 0 displaystyle b neq 0 nbsp lim x x sin 1 x 1 displaystyle lim x to infty x sin left frac 1 x right 1 nbsp lim x 0 1 cos x x 0 displaystyle lim x to 0 frac 1 cos x x 0 nbsp lim x 0 1 cos x x 2 1 2 displaystyle lim x to 0 frac 1 cos x x 2 frac 1 2 nbsp lim x n tg p x p 2 displaystyle lim x to n pm operatorname tg left pi x frac pi 2 right mp infty nbsp dlya vsih n Z displaystyle n in mathbb Z nbsp lim n sin sin sin x 0 n 0 displaystyle lim n to infty underbrace sin sin cdots sin x 0 n 0 nbsp de x 0 R displaystyle x 0 in mathbb R nbsp dovilne lim n cos cos cos x 0 n d displaystyle lim n to infty underbrace cos cos cdots cos x 0 n d nbsp de d displaystyle d nbsp chislo Dotti en x 0 R displaystyle x 0 in mathbb R nbsp dovilne Dzherela RedaguvatiGrigorij Mihajlovich Fihtengolc Kurs diferencialnogo ta integralnogo chislennya 2023 1300 s ukr Primitki Redaguvati Some Special Limits www sosmath com SOME IMPORTANT LIMITS Math Formulas Mathematics Formulas Basic Math Formulas www pioneermathematics com Arhiv originalu za 31 lipnya 2019 Procitovano 25 chervnya 2022 Otrimano z https uk wikipedia org w index php title Spisok granic amp oldid 40597820
