Теория пределов и числовые ряды (1092163), страница 5
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Ïðè x → ∞ çíàìåíàòåëü âûðàx→0 xæåíèÿ åñòü áåñêîíå÷íî áîëüøàÿ âåëè÷èíà ( lim x = ∞), à ÷èñëèòåëüx→∞ îãðàíè÷åííàÿ (| sin x| 6 1), è îòíîøåíèå îãðàíè÷åííîé âåëè÷èíû êáåñêîíå÷íî áîëüøîé åñòü áåñêîíå÷íî ìàëàÿ âåëè÷èíà. Ñëåäîâàòåëüíî,sin x= 0.x→∞ xlimÏðèìåð 7.11.Íàéòè limÐ å ø å í è å: Ïðèx.x→0 tg 5xx → 0 èìååì äåëî ñ íåîïðåäåë¼ííîñòüþ òèïà0.0Ñîâåðøèì ýëåìåíòàðíûå ïðåîáðàçîâàíèÿlim cos 5xxxcos 5x= lim sin 5x = lim sin 5x = x→0 sin 5x .x→0 tg 5xx→0x→0lim xcos 5xxlimx→0Èçâåñòíî, ÷òîlim cos 5x = 1, à â ïðèìåðå 7.8 ïîêàçàíî, ÷òî limx→0ñëåäîâàòåëüíî,x→0sin 5xx= 5,x1= .x→0 tg 5x5limÏðè ðåøåíèè ïðèìåðîâ, ñâÿçàííûõ ñ íàõîæäåíèåì ïðåäåëîâ îòòðèãîíîìåòðè÷åñêèõ ôóíêöèé, â íåêîòîðûõ ñëó÷àÿõ ðåêîìåíäóåòñÿÏðàêòè÷åñêîå çàíÿòèå 7. Òåîðèÿ ïðåäåëîâ129âîñïîëüçîâàòüñÿ òðèãîíîìåòðè÷åñêèìè òîæäåñòâàìè, ÷òîáû çàòåì ëèáî âû÷èñëèòü ïðåäåë íåïîñðåäñòâåííîé ïîäñòàíîâêîé ïðåäåëüíîãî çíà÷åíèÿ àðãóìåíòà, ëèáî ïðèâåñòè ïîëó÷åííûé ïðåäåë ê ïåðâîìó çàìåsin x.÷àòåëüíîìó ïðåäåëó limx→0 xÍàéòè limÏðèìåð 7.12.x→0x → 0 èìååò ìåñòî íåîïðåäåë¼ííîñòü âèäà 00 ,lim (1 − cos 5x) = lim 1 − lim cos 5x = 1 − 1 = 0.
ÂîñïîëüçóåìñÿÐ å ø å í è å:òàê êàê1−cos 5x.x2Ïðèx→0x→0x→0òðèãîíîìåòðè÷åñêèì òîæäåñòâîìïðèìåð â âèäå:1 − cos 5x = 2 sin22 sin21 − cos 5x=limx→0x→0x2x2lim5x25xè ïåðåïèøåì2.Âûïîëíèì íåîáõîäèìûå ïðåîáðàçîâàíèÿ äëÿ òîãî, ÷òîáû ïðèâåñòè ïîëó÷åííûé ïðåäåë ê âèäó ïåðâîãî çàìå÷àòåëüíîãî ïðåäåëà. Îáîçíà÷èì5x= y , îòêóäà x = 2y. Î÷åâèäíî, ÷òî ïðè x → 0 è y → 0 :252 sin2 5x252 sin2 y2 sin2 ysin2 y2=lim=lim=limlim=()4 22y 2x→0y→0y→0x22 y→0 y 2y255()2=sin yy25lim2 y→0=2525·1= .221−sin x2.x→π π−xÍàéòè limÏðèìåð 7.13.Ïðè ïîäñòàíîâêå x =1 − sin x21 − sin π2Ð å ø å í è å:limx→ππ−x=π−ππïîä çíàê ïðåäåëà ïîëó÷èì=1−10= ,π−π0ò.å. íåîïðåäåë¼ííîñòü.
Äëÿ ðàñêðûòèÿ íåîïðåäåë¼ííîñòè îáîçíà÷èìx − π = y, ay → 0.çíà÷èòÏðèx → π, x − π → 0è, ñëåäîâàòåëüíî,(y π )xyy+π= cos ,= sin= sin+222 221 − cos y21 − sin x2= lim.limx→π π − xy→0−ysinÍåîïðåäåë¼ííîñòüâàòüñÿ ôîðìóëîéx = y + π.10ñîõðàíèëàñü, íî òåïåðü ìû ñìîæåì âîñïîëüçî0− cos y2 = 2 sin2 y4 .2 sin2 y41 − cos y2= lim.limy→0y→0−y−y130Ïðàêòè÷åñêîå çàíÿòèå 7. Òåîðèÿ ïðåäåëîâÎáîçíà÷èìy4= z,îòêóäày = 4z .Ïðèy → 0, z → 0 :2 sin2 y42 sin2 z1sin2 z= lim= − lim=y→0z→0 −4z−y2 z→0 z11sin z= − lim· lim sin z = − · 1 · 0 = 0.z→02 z→0 z21−sin x2Ñëåäîâàòåëüíî, lim= 0.π−xlimx→πÏðèìåð 7.14.Ð å ø å í è å:íåîïðåäåë¼ííîñòüÍàéòè lim (cos(mx) − cos(nx))/x2 .x→0Ïðè ïîäñòàíîâêå x=0 ïîä çíàê ïðåäåëà ïîëó÷aeì1−1= 00 .
Ïðèìåíèì òðèãoíîìåòðè÷åñêîå òîæäåñòâî0cos(mx) −)()(− cos(nx) = −2 sin m+nx · sin m−nx :22()( m−n )−2 sin m+nx·sinxcos(mx) − cos(nx)22lim=lim=x→0x→0x2x2( m+n )( m−n )sin 2 xsin 2 x= −2 lim· lim=x→0x→0xx (())m+nm+nm−nm−n·sinx·sinx= −2 lim 2 m+n 2· lim 2 m−n 2=x→0x→0·x·x22()()sin m+nx m−nsin m−nxm+n22· lim m+n·· lim m−n== −2 ·x→0x→02·x2·x222n 2 − m2= − (m2 − n2 ) · 1 · 1 =.42Ïðèìåð 7.15.()xÍàéòè lim 1 + x3 .x→∞Ð å ø å í è å: Ïî âíåøíåìó âèäó äàííûé ïðèìåð íàïîìèíàåò âòî33ðîé çàìå÷àòåëüíûé ïðåäåë. Ââåäåì ïåðåìåííóþ t = , îòñþäà x = .xt3Ïðè x → ∞ t =→ 0.x()x()1 3333ttlimx→∞1+x= lim(1 + t) = lim(1 + t)t→0t→0=e .Ïðàêòè÷åñêîå çàíÿòèå 7. Òåîðèÿ ïðåäåëîâÏðèìåð 7.16.Íàéòè limx→∞( x−1 )xx+1131.Ð å ø å í è å: Âîñïîëüçóåìñÿ ñâîéñòâîì ïðåäåëà ôóíêöèè è ñîâåðøèì ñëåäóþùèå ýëåìåíòàðíûå ïðåîáðàçîâàíèÿ, ðàçäåëèâ ÷èñëèòåëü èçíàìåíàòåëü íà(limx→∞x−1x+1)xx:(= limÏðèìåð 7.17.x→∞x · (1 −x · (1 +Íàéòè limx→∞( 2x+3 )x2x−1)x(lim 1 + −1xe−1)== e−2 .= x→∞ (xelim 1 + x1x→∞.2x − 1 = y , îòêóäà x =y+1,22x + 3 = y + 4.x → ∞ è y → ∞.()x() y+1()( y + 1 )2x + 3y+4 24 2 2lim= lim= lim 1 +=x→∞y→∞y→∞2x − 1yy)y)1 ()y ) 12(((14 24 24= lim 1 +· lim 1 += lim 1 +· 12 =y→∞y→∞y→∞yyyÐ å ø å í è å: Îáîçíà÷èìÏðè1 )x)x1)x1= (e4 ) 2 · 1 = e2 .Ïðèìåð 7.18.Íàéòè lim (ln(2x + 1) − ln(x + 2)).x→∞Ïðè x → ∞ èìååì äåëî ñ íåîïðåäåë¼ííîñòüþ∞ − ∞.
Âîñïîëüçóåìñÿ ñâîéñòâîì ëîãàðèôìè÷åñêîé ôóíêöèèln(2x + 1) − ln(x ++ 2) = ln 2x+1. ×èñëèòåëü è çíàìåíàòåëü ïîäëîãàðèôìè÷åñêîãî âûx+2Ð å ø å í è å:âèäàðàæåíèÿ ðàçäåëèì íà2+ 1x ⇒ ln 2x+1= ln 1+ x2x+2è ïîäñòàâèì â èñõîäíûéxïðèìåð:(lim (ln(2x + 1) − ln(x + 2)) = limx→∞òàê êàê ïðèx→∞x→∞12èðàâíû íóëþ.xx2+ln1+1x2x)= ln 2,132Ïðàêòè÷åñêîå çàíÿòèå 7. Òåîðèÿ ïðåäåëîâÑàìîñòîÿòåëüíàÿ ðàáîòàÄëÿ ïðèâåä¼ííûõ íèæå ïîñëåäîâàòåëüíîñòåé çàïèñàòü ôîðìóëóîáùåãî ÷ëåíà ïîñëåäîâàòåëüíîñòè.Ïðèìåð 7.19.1, 12 , 212 , 213 , · · · .Ïðèìåð 7.20.2, 32 , 43 , 45 , 65 , · · · .Ïðèìåð 7.21.1, 4, 9, 16, 25, · · · .2 4 6 8, , , ,··· .1 3 5 71 11Ïðèìåð 7.23.
1, − , , − , · · ·2 34Ïðèìåð 7.22..Ïîñëåäîâàòåëüíîñòü yn çàäàíà ôîðìóëîé îáùåãî÷ëåíà ïîñëåäîâàòåëüíîñòè {yn } = 3nn Íàéòè y3 , y5 , yn+1 .Ïðèìåð 7.24.Íàéòè ïðåäåëû ïîñëåäîâàòåëüíîñòåéÏðèìåð 7.25.1, 6; 1, 66; 1, 666; 1, 6666; . . .Ïðèìåð 7.26.[limn→∞Ïðèìåð 7.27.]1 + 3 + 5 + 7 + ... + (2n − 1).n2]1(−1)n−11 1lim 1 − + −+ ··· +.n→∞3 9 273n−1Ïðèìåð 7.28.[9n + 8n.n→∞ 9n+1 + 8n+1√√√√√√2,2,2, . .
. .limÏðèìåð 7.29.Ïðèìåð 7.30.n sin n!.n→∞ n2 + 1limÍàéòè îäíîñòîðîííèå ïðåäåëû:Ïðèìåð 7.31.Ïðèìåð 7.32.1lim 2 n−4 .n→4−01lim 2 n−4 .n→4+0Ëåêöèÿ 8. Ïðè¼ìû ðàñêðûòèÿ íåîïðåäåë¼ííîñòåé133Íàéòè ïðåäåëû ôóíêöèé.Ïðèìåð 7.33.Ïðèìåð 7.34.Ïðèìåð 7.35.Ïðèìåð 7.36.Ïðèìåð 7.37.Ïðèìåð 7.38.Ïðèìåð 7.45.sin 4x.xÏðèìåð 7.39.lim sin 5x .x→0 sin 2xÏðèìåð 7.40.sin x.x→2 xÏðèìåð 7.41.sin x.x→0 tg xÏðèìåð 7.42.limx→0limlimsin2 x32 .x→0 xlimlimx→0tg x−sin x.x3lim cos x .x→ π2 π−2xx→asin x−sin a.x−alim.lim(x→∞lim)xxx+1( 2+x )x3−xx→0(Ïðèìåð 7.43.Ïðèìåð 7.44.limx→∞limx→0.x2 +22x2 +1)x 2.ln(1−3x).xlim x (ln(x + 1) − ln x) .x→∞Ëåêöèÿ 8. Ïðè¼ìû ðàñêðûòèÿ íåîïðåäåë¼ííîñòåéÑðàâíåíèå áåñêîíå÷íî ìàëûõ ôóíêöèé.
Ýêâèâàëåíòíîñòü áåñêîíå÷íî ìàëûõ ôóíêöèé. Ïðèåìû ðàñêðûòèÿ íåîïðåäåë¼ííîñòåé.8.1. Ñðàâíåíèå áåñêîíå÷íî ìàëûõ ôóíêöèéÔóíêöèè α(x) è β(x) íàçûâàþòñÿ áåñêîíå÷íîìàëûìè îäíîãî ïîðÿäêà ìàëîñòè ïðè x → a, åñëè lim α(x)= b ̸= 0x→a β(x)è ̸= ∞.Îïðåäåëåíèå 8.1.Îïðåäåëåíèå 8.2. Ôóíêöèÿ α(x) íàçûâàåòñÿ áåñêîíå÷íî ìàëîéáîëåå âûñîêîãî ïîðÿäêà ìàëîñòè, ÷åì ôóíêöèÿ β(x) ïðè x → a, åñëèlim α(x)= 0.β(x)x→aÔóíêöèÿ α(x) íàçûâàåòñÿ áåñêîíå÷íî ìàëîéáîëåå íèçêîãî ïîðÿäêà ìàëîñòè, ÷åì ôóíêöèÿ β(x) ïðè x → a, åñëèlim α(x)= ∞.β(x)Îïðåäåëåíèå 8.3.x→aÔóíêöèè α(x) è β(x) íàçûâàþòñÿ íåñðàâíèíå ñóùåñòâóåò èìûìè áåñêîíå÷íî ìàëûìè ïðè x → a, åñëè lim α(x)x→a β(x)íå ðàâåí ∞.Îïðåäåëåíèå 8.4.134Ëåêöèÿ 8.
Ïðè¼ìû ðàñêðûòèÿ íåîïðåäåë¼ííîñòåéÏðèìåð 8.1.Ñðàâíèòü áåñêîíå÷íî ìàëûå ôóíêöèèy = x2 è y = 3x ïðè x → 0.Ð å ø å í è å:x2x0= lim = = 0.x→0 3xx→0 332Ñëåäîâàòåëüíî, ôóíêöèÿ y = x áåñêîíå÷íî ìàëàÿâûñîêîãî ïîðÿäêà ìàëîñòè, ÷åì ôóíêöèÿ y = 3x.limÏðèìåð 8.2.ïðèx→0áîëååÑðàâíèòü áåñêîíå÷íî ìàëûå ôóíêöèèy = x2 + x − 6 è y = 4 − x2 ïðè x → 2.Ð å ø å í è å:5x2 + x − 6(x − 2)(x + 3)x+3= lim= − lim= − ̸= 0.2x→2x→2 −(x − 2)(x + 2)x→2 x + 24−x4limÑëåäîâàòåëüíî, óêàçàííûå ôóíêöèè ÿâëÿþòñÿ áåñêîíå÷íî ìàëûìè îäíîãî ïîðÿäêà ìàëîñòè ïðèÏðèìåð 8.3.x → 2.Ñðàâíèòü áåñêîíå÷íî ìàëûå ôóíêöèèy=Ð å ø å í è å:cos x1è y = ïðè x → ∞.xxcos xxx→∞ 1xlimÒàê êàêcos x= lim cos x.x→∞íå èìååò ïðåäåëà ïðèx → ∞,óêàçàííûå ôóíêöèè ÿâ-ëÿþòñÿ íåñðàâíèìûìè áåñêîíå÷íî ìàëûìè ïðèx → ∞.8.2.
Ýêâèâàëåíòíîñòü áåñêîíå÷íî ìàëûõ ôóíêöèéÔóíêöèè α(x) è β(x), áåñêîíå÷íî ìàëûå ïðèx → a, íàçûâàþòñÿ ýêâèâàëåíòíûìè (ðàâíîñèëüíûìè), åñëè ïðåäåëèõ îòíîøåíèÿ lim α(x)= 1.β(x)Îïðåäåëåíèå 8.5.x→aÒîãäà äëÿ çíà÷åíèé x, áëèçêèõ ê x = a, èìååò ìåñòî ïðèáëèæ¼ííîåα(x)ðàâåíñòâî lim≈ 1, èëè α(x) ≈ β(x), òî÷íîñòü êîòîðîãî âîçðàñòàåòx→a β(x)x ê a.Åñëè α(x) è β(x)ýêâèâàëåíòíûåx → a, òî ïèøóò α(x) ∼ β(x).ñ ïðèáëèæåíèåììàëûå ïðèáåñêîíå÷íîËåêöèÿ 8. Ïðè¼ìû ðàñêðûòèÿ íåîïðåäåë¼ííîñòåé135Òåîðåìà 8.1. Ïðåäåë îòíîøåíèÿ äâóõ áåñêîíå÷íî ìàëûõ ôóíêöèé ðàâåí ïðåäåëó îòíîøåíèÿ ýêâèâàëåíòíûõ èì ôóíêöèé, ò. å. åñëèα ∼ α1 , a β ∼ β1 ïðè x → a, òîlimx→aÄàíî:α ∼ α1 ,aβ ∼ β1α1 (x)α(x)= lim.x→aβ1 (x)β(x)ïðèx → a.Äîêàçàòåëüñòâî:α(x)α(x) α1 (x) β1 (x)= lim··=x→a β(x)x→a β(x) α1 (x) β1 (x)lim= limx→aα(x) α1 (x) β1 (x)α1 (x)α1 (x)··=1·· 1 = lim.x→aα1 (x) β1 (x) β(x)β1 (x)β1 (x)Ïðèìåð 8.4.Íàéòè limÐ å ø å í è å:sin 3x ∼ 3xïðèsin 5x.x→ 0 sin 3xÒàê êàêx → 0,limx→0sin 5x5x= 1, limx→0sin 3x3x= 1,òîsin 5x ∼ 5x,èsin 5x5x5= lim= .x→0 sin 3xx→0 3x3limÁåñêîíå÷íî ìàëûå ôóíêöèè α(x) è β(x) ýêâèâàëåíòíû, åñëè èõ ðàçíîñòü [α(x) − β(x)] åñòü áåñêîíå÷íî ìàëàÿ ôóíêöèÿ áîëåå âûñîêîãî ïîðÿäêà ìàëîñòè, ÷åì α(x) è β(x).Òåîðåìà 8.2.Äàíî:lim α(x) = lim β(x) = 0x→aìàëûå ïðè x →Äîêàçàòü, ÷òîx→a ôóíêöèèα(x)a è γ(x) = α(x) − β(x).γ(x)= limα(x) ∼ β(x) ò.
å. lim α(x)x→aγ(x)x→a β(x)èβ(x)áåñêîíå÷íî= 0.Äîêàçàòåëüñòâî()β(x)(φ(x) − ψ(x))ψ(x)lim= lim= lim 1 −=x→a φ(x)x→ax→aφ(x)φ(x)ψ(x)= 1 − 1 = 0.x→a φ(x)= 1 − limÑëåäîâàòåëüíî,γ(x)åñòü áåñêîíå÷íî ìàëàÿ ôóíêöèÿ áîëåå âûñîêîãîγ(x)ïîðÿäêà ìàëîñòè, ÷åì α(x). Àíàëîãè÷íî ìîæíî äîêàçàòü, ÷òî lim=x→a β(x)0.136Ëåêöèÿ 8. Ïðè¼ìû ðàñêðûòèÿ íåîïðåäåë¼ííîñòåéÒåîðåìà 8.3. Ñóììà êîíå÷íîãî ÷èñëà áåñêîíå÷íî ìàëûõ ôóíêöèéðàçëè÷íûõ ïîðÿäêîâ ýêâèâàëåíòíà ñëàãàåìîìó íèçøåãî ïîðÿäêà.Äàíî:lim γ(x) = lim α(x) = lim β(x) = 0x→ax→ax→aáåñêîíå÷íî ìàëûå ïðèγ(x), α(x), β(x)ñòè γ(x) áåñêîíå÷íî ôóíêöèèx → a.Ïóñòü äëÿ îïðåäåë¼ííî-ìàëàÿ ôóíêöèÿ íèçøåãî ïîðÿäêà ìàëîñòè ïîα(x)ñðàâíåíèÿ ñ îñòàëüíûìè ñëàãàåìûìè, ò.å.
lim= lim β(x)= 0.γ(x)x→ax→a γ(x)γ(x)+α(x)+β(x)= 1, ò.å. ñóììà áåñêîíå÷íî ìàëûõγ(x)x→a→ a ýêâèâàëåíòíà â äàííîì ñëó÷àå γ(x).Äîêàçàòü, ÷òîôóíêöèé ïðèxlimÄîêàçàòåëüñòâîγ(x) + α(x) + β(x)α(x)β(x)= lim 1 + lim+ lim= 1 + 0 + 0 = 1.x→ax→ax→a γ(x)x→a γ(x)γ(x)limÏðèìåð 8.5.5x+6x2.x→0 sin 2xÍàéòè limx → 0 5x + 6x2 ∼ 5x (ïî5x+6x2òî lim= lim 5x= 52 .sin 2x2xÐ å ø å í è å: Òàê êàê ïðèsin 2x ∼ 2x(ïî òåîðåìå 8.1),x→0òåîðåìå 8.3) èx→08.3. Ïðèåìû ðàñêðûòèÿ íåîïðåäåë¼ííîñòåéÏðè ðàññìîòðåíèè àðèôìåòè÷åñêèõ îïåðàöèé íàä ïðåäåëàìè ïðåäïîëàãàåòñÿ, ÷òî îáå ïåðåìåííûå âåëè÷èíû èìåþò ïðåäåëx → x0 ,à âñëó÷àå ïðåäåëà ÷àñòíîãî îãîâàðèâàåòñÿ, ÷òî ïðåäåë çíàìåíàòåëÿ íåðàâåí íóëþ.Ñóùåñòâóþò ñëó÷àè, êîãäà ýòè óñëîâèÿ íå âûïîëíÿþòñÿ.
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