Stiskna teorema teorema pro dvoh policejskih angl squeeze theorem teorema v matematichnomu analizi pro granicyu funkciyi yaka zatisnuta mizh dvoma inshimi funkciyami sho mayut rivni granici Vizualizaciya teoremi pro stisnennya dlya funkcij g x x 2 displaystyle color OliveGreen g x x 2 f x x 2 sin 1 x displaystyle color Blue f x x 2 sin frac 1 x ta h x x 2 displaystyle color Orange h x x 2 pri x 0 displaystyle x to 0 Teorema pro stisnennya takozh vidoma pid nazvami yak teorema pro dvoh policejskih teorema pro dvoh karabineriv i teorema pro dvoh zhandarmiv Interpretaciya polyagaye v tomu sho yaksho dvoye policejskih suprovodzhuyut p yanogo uv yaznenogo mizh soboyu i obidva oficeri jdut do kameri to nezalezhno vid togo yak kolivayetsya uv yaznenij vin vse odno opinitsya v kameri Zmist 1 Formulyuvannya 2 Dovedennya 3 Priklad 3 1 Pershij priklad 3 2 Drugij priklad 3 3 Tretij priklad 3 4 Chetvertij priklad 4 Teorema pro tri poslidovnosti 4 1 Dovedennya 5 Literatura 6 PosilannyaFormulyuvannya RedaguvatiNehaj A R displaystyle A subset mathbb R nbsp x 0 displaystyle x 0 nbsp granichna tochka mnozhini A displaystyle A nbsp funkciyi f g i h viznacheni na A displaystyle A nbsp dlya yakih vikonuyutsya nastupni umovi g x f x h x displaystyle g x leqslant f x leqslant h x nbsp dlya vsih x A displaystyle x in A nbsp lim x x 0 g x lim x x 0 h x a displaystyle lim x to x 0 g x lim x to x 0 h x a nbsp Todi lim x x 0 f x a displaystyle lim x to x 0 f x a nbsp Funkciyi g i h nazivayut verhnoyu ta nizhnoyu mezhami f vidpovidno Tut x 0 displaystyle x 0 nbsp ne musit buti vnutrishnoyu tochkoyu mnozhini A displaystyle A nbsp Shozhe tverdzhennya spravedlive i pri x displaystyle x to infty nbsp abo x displaystyle x to infty nbsp Dovedennya RedaguvatiProvedemo dovedennya iz vikoristannyam oznachennya granici funkciyi v tochci za Koshi tobto nam potribno dovesti sho dlya kozhnogo dijsnogo e gt 0 displaystyle varepsilon gt 0 nbsp isnuye dijsne d gt 0 displaystyle delta gt 0 nbsp take sho dlya vsih x A x 0 d x 0 d x 0 displaystyle x in A cap x 0 delta x 0 delta setminus x 0 nbsp vikonuyetsya f x a lt e displaystyle f x a lt varepsilon nbsp Tobto e gt 0 d gt 0 x A x 0 d x 0 d x 0 f x a lt e displaystyle forall varepsilon gt 0 exists delta gt 0 forall x in A cap x 0 delta x 0 delta setminus x 0 f x a lt varepsilon nbsp Z togo sho lim x x 0 g x a displaystyle lim x to x 0 g x a nbsp viplivaye sho e gt 0 d 1 gt 0 x A x 0 d 1 x 0 d 1 x 0 g x a lt e displaystyle forall varepsilon gt 0 exists delta 1 gt 0 forall x in A cap x 0 delta 1 x 0 delta 1 setminus x 0 g x a lt varepsilon nbsp 1 i z togo sho lim x x 0 h x a displaystyle lim x to x 0 h x a nbsp viplivaye sho e gt 0 d 2 gt 0 x A x 0 d 2 x 0 d 2 x 0 h x a lt e displaystyle forall varepsilon gt 0 exists delta 2 gt 0 forall x in A cap x 0 delta 2 x 0 delta 2 setminus x 0 h x a lt varepsilon nbsp 2 Takozh mayemo sho g x f x h x displaystyle g x leqslant f x leqslant h x nbsp zvidsi g x a f x a h x a displaystyle g x a leqslant f x a leqslant h x a nbsp x A displaystyle forall x in A nbsp Poklademo d min d 1 d 2 displaystyle delta min delta 1 delta 2 nbsp Todi dlya vsih x A x 0 d x 0 d x 0 displaystyle x in A cap x 0 delta x 0 delta setminus x 0 nbsp poyednavshi 1 ta 2 otrimayemo e lt g x a f x a h x a lt e displaystyle varepsilon lt g x a leqslant f x a leqslant h x a lt varepsilon nbsp e lt f x a lt e displaystyle varepsilon lt f x a lt varepsilon nbsp sho j treba bulo dovesti Priklad RedaguvatiPershij priklad Redaguvati nbsp x2 sin 1 x zatisnuta yak x pryamuye do 0Granicyu lim x 0 x 2 sin 1 x displaystyle lim x to 0 x 2 sin tfrac 1 x nbsp nemozhlivo vstanoviti cherez zakon lim x a f x g x lim x a f x lim x a g x displaystyle lim x to a f x cdot g x lim x to a f x cdot lim x to a g x nbsp bo lim x 0 sin 1 x displaystyle lim x to 0 sin tfrac 1 x nbsp ne isnuye Odnak z viznachennya sinusa 1 sin 1 x 1 displaystyle 1 leq sin tfrac 1 x leq 1 nbsp Viplivaye sho x 2 x 2 sin 1 x x 2 displaystyle x 2 leq x 2 sin tfrac 1 x leq x 2 nbsp Z togo sho lim x 0 x 2 lim x 0 x 2 0 displaystyle lim x to 0 x 2 lim x to 0 x 2 0 nbsp za stisknoyu teoremoyu lim x 0 x 2 sin 1 x displaystyle lim x to 0 x 2 sin tfrac 1 x nbsp povinen buti 0 Drugij priklad Redaguvati nbsp Porivnyannya plosh A A D F A sektor A D B A A D B 1 2 tg x 1 x 2 p p 1 2 sin x 1 sin x cos x x sin x cos x sin x 1 x 1 sin x cos x sin x x 1 displaystyle begin aligned amp A triangle ADF geqslant A text sektor ADB geqslant A triangle ADB Rightarrow amp frac 1 2 cdot text tg x cdot 1 geqslant frac x 2 pi cdot pi geq frac 1 2 cdot sin x cdot 1 Rightarrow amp frac sin x cos x geqslant x geqslant sin x Rightarrow amp frac cos x sin x leqslant frac 1 x leqslant frac 1 sin x Rightarrow amp cos x leqslant frac sin x x leqslant 1 end aligned nbsp Napevno najvidomishi prikladi znahodzhennya granic cherez teoremu zatiskannya ce dovedennya togo sho lim x 0 sin x x 1 lim x 0 1 cos x x 0 displaystyle begin aligned amp lim x to 0 frac sin x x 1 10pt amp lim x to 0 frac 1 cos x x 0 end aligned nbsp Persha granicya viplivaye z vikoristannya stisknoyi teoremi i togo faktu sho cos x sin x x 1 displaystyle cos x leqslant frac sin x x leqslant 1 nbsp dlya x dosit blizkogo ale ne rivnogo 0 Pravilnist yakogo dlya dodatnogo x mozhna pobachiti za dopomogoyu geometrichnih mirkuvan div risunok yaki takozh mozhna poshiriti na vid yemne x Chasto cyu granicyu nazivayut pershoyu chudovoyu graniceyu Druga viplivaye z teoremi stisnennya i togo faktu sho 0 1 cos x x x displaystyle 0 leqslant frac 1 cos x x leqslant x nbsp dlya x dosit blizkogo ale ne rivnogo 0 Ce mozhna otrimati zaminivshi sin x displaystyle sin x nbsp u poperednomu fakti na 1 cos 2 x displaystyle sqrt 1 cos 2 x nbsp i pidnisshi otrimanu nerivnist do kvadratu Ci dvi granici vikoristovuyutsya dlya dovedennya togo sho pohidna sinusa ye kosinus Na ce spirayutsya vivedennya pohidnih dlya inshih trigonometrichnih funkcij Tretij priklad Redaguvati Mozhna pokazati sho tg 8 1 cos 2 8 sec 2 8 displaystyle text tg theta prime frac 1 cos 2 theta sec 2 theta nbsp nbsp Na risunku plosha menshogo z dvoh zashtrihovanih sektoriv kruga dorivnyuye D 8 sec 2 8 2 D 8 2 cos 2 8 displaystyle frac Delta theta sec 2 theta 2 frac Delta theta 2 cos 2 theta nbsp oskilki radius dorivnyuye 1 cos 8 displaystyle frac 1 cos theta nbsp a duga na odinichnomu koli maye dovzhinu D8 Analogichno plosha bilshogo z dvoh zashtrihovanih sektoriv dorivnyuye D 8 sec 2 8 D 8 2 D 8 2 cos 2 8 D 8 displaystyle frac Delta theta sec 2 theta Delta theta 2 frac Delta theta 2 cos 2 theta Delta theta nbsp Mizh sektorami stisnutij trikutnik osnovoyu yakogo ye vertikalnij vidrizok yakij spoluchaye dvi vidileni na risunku tochki Dovzhina osnovi trikutnika dorivnyuye tg 8 D 8 tg 8 displaystyle text tg theta Delta theta text tg theta nbsp a visota 1 displaystyle 1 nbsp Otzhe plosha trikutnika dorivnyuye tg 8 D 8 tg 8 2 displaystyle frac text tg theta Delta theta text tg theta 2 nbsp Z nerivnostej D 8 2 cos 2 8 tg 8 D 8 tg 8 2 D 8 2 cos 2 8 D 8 displaystyle frac Delta theta 2 cos 2 theta leqslant frac text tg theta Delta theta text tg theta 2 leqslant frac Delta theta 2 cos 2 theta Delta theta nbsp viplivaye 1 cos 2 8 tg 8 D 8 tg 8 D 8 1 cos 2 8 D 8 displaystyle frac 1 cos 2 theta leqslant frac text tg theta Delta theta text tg theta Delta theta leqslant frac 1 cos 2 theta Delta theta nbsp za umovi D8 gt 0 a yaksho yaksho D8 lt 0 to nerivnist perevertayetsya Oskilki pershij i tretij virazi pryamuyut do 1 cos 2 8 displaystyle frac 1 cos 2 theta nbsp pri D 8 0 displaystyle Delta theta to 0 nbsp a serednij viraz pryamuye do tg 8 displaystyle text tg theta prime nbsp to ce daye bazhanij rezultat Chetvertij priklad Redaguvati Stiskna teorema takozh mozhe vikoristovuvatisya v bagatovimirnomu analizi U comu vipadku funkciyi g ta h povinni obmezhuvati f v okoli tochki sho cikavit i vona pracyuye lishe todi yaksho funkciya f dijsno tam maye granicyu Takim chinom u bagatovimirnomu vipadku cyu teoremu mozhna vikoristovuvati shob dovesti sho funkciya f maye granicyu v tochci ale yiyi ne mozhna vikoristovuvati shob dovesti vidsutnist granici v tochci lim x y 0 0 x 2 y x 2 y 2 displaystyle lim x y to 0 0 frac x 2 y x 2 y 2 nbsp ne mozhna znajti vzyavshi bud yaki granici uzdovzh krivih yaki prohodyat cherez tochku 0 0 displaystyle 0 0 nbsp ale oskilki 0 x 2 x 2 y 2 1 displaystyle 0 leqslant frac x 2 x 2 y 2 leqslant 1 nbsp y y y displaystyle left y right vert leqslant y leqslant left y right vert nbsp y x 2 y x 2 y 2 y displaystyle left y right vert leqslant frac x 2 y x 2 y 2 leqslant left y right vert nbsp lim x y 0 0 y 0 displaystyle lim x y to 0 0 left y right vert 0 nbsp lim x y 0 0 y 0 displaystyle lim x y to 0 0 left y right vert 0 nbsp 0 lim x y 0 0 x 2 y x 2 y 2 0 displaystyle 0 leqslant lim x y to 0 0 frac x 2 y x 2 y 2 leqslant 0 nbsp to za stisknoyu teoremoyu lim x y 0 0 x 2 y x 2 y 2 0 displaystyle lim x y to 0 0 frac x 2 y x 2 y 2 0 nbsp Teorema pro tri poslidovnosti Redaguvati nbsp Koli poslidovnist lezhit mizh dvoma inshimi zbizhnimi poslidovnostyami z takoyu zh graniceyu to vona takozh shodit do ciyeyi graniciStiskna teorema takozh spravedliva dlya poslidovnostej yak funkcij cilogo argumentu Nehaj dlya poslidovnostej a n n 1 displaystyle a n n 1 infty nbsp b n n 1 displaystyle b n n 1 infty nbsp i c n n 1 displaystyle c n n 1 infty nbsp vikonuyutsya nastupni umovi n 1 a n b n c n displaystyle forall n geqslant 1 a n leqslant b n leqslant c n nbsp lim n a n lim n c n l displaystyle lim n to infty a n lim n to infty c n l nbsp Todi lim n b n l displaystyle lim n to infty b n l nbsp U comu vipadku yiyi chasto nazivayut teoremoyu pro tri poslidovnosti Dovedennya shozhe yak i dlya funkcij dijsnogo argumentu Dovedennya Redaguvati Nehaj e gt 0 displaystyle varepsilon gt 0 nbsp zadane Zgidno z umovi teoremi n 1 N n n 1 l e lt a n displaystyle exists n 1 in mathbb N forall n geqslant n 1 l varepsilon lt a n nbsp n 2 N n n 2 c n lt l e displaystyle exists n 2 in mathbb N forall n geqslant n 2 c n lt l varepsilon nbsp Todi dlya vsih n n 0 max n 1 n 2 displaystyle n geqslant n 0 max n 1 n 2 nbsp mayemo l e lt a n b n c n lt l e displaystyle l varepsilon lt a n leqslant b n leqslant c n lt l varepsilon nbsp zvidki viplivaye sho l e lt b n lt l e displaystyle l varepsilon lt b n lt l varepsilon nbsp sho i treba bulo dovesti Literatura RedaguvatiDorogovcev A Ya Matematichnij analiz Pidruchnik U dvoh chastinah Chastina 1 Kiyiv Libid 1993 320 s ISBN 5 325 00380 1 Dorogovcev A Ya Matematichnij analiz Pidruchnik U dvoh chastinah Chastina 2 Kiyiv Libid 1994 304 s ISBN 5 325 00351 X Posilannya RedaguvatiWeisstein Eric W Stiskna teorema angl na sajti Wolfram MathWorld Otrimano z https uk wikipedia org w index php title Stiskna teorema amp oldid 40219801
