Теория пределов и числовые ряды (Конспект), страница 6

Íàïðèìåð, ïåðåìåííûå, ñòîÿùèå â ÷èñëèòåëå è çíàìåíàòåëå, ïðèx → x0ñòðåìÿòñÿ îäíîâðåìåííî ê íóëþ èëè áåñêîíå÷íîñòè.  ýòîì ñëó÷àå ãî0âîðÿò, ÷òî èìååò ìåñòî íåîïðåäåë¼ííîñòü ñîîòâåòñòâóþùåãî òèïà:0∞èëè.∞Åñëè ñóììà áåñêîíå÷íî áîëüøèõ âåëè÷èí ïðè x → x0 îäíîãî çíàêàåñòü âåëè÷èíà áåñêîíå÷íî áîëüøàÿ, òî î ïðåäåëå ðàçíîñòè òàêèõ âåëè÷èí çàðàíåå íè÷åãî ñêàçàòü íåëüçÿ íåîïðåäåë¼ííîñòü òèïàÏðè óìíîæåíèè áåñêîíå÷íî ìàëîé âåëè÷èíû ïðèíå÷íî áîëüøóþ ïðèx → x0x → x0∞ − ∞.íà áåñêî-âîçíèêàåò íåîïðåäåë¼ííîñòü òèïà0 · ∞.Ïðè ðàññìîòðåíèè 2-ãî çàìå÷àòåëüíîãî ïðåäåëà ìû òàêæå âèäåëè,1÷òîx→x0ãäåα(x)èβ(x)áåñêîíå÷íî ìàëûå âåëè÷èíûx → x0 , ìîæåò áûòü ðàçëè÷íûì, âñå çàâèñèòβ(x) ê íóëþ ïðè x → x0 . Ýòî íåîïðåäåë¼ííîñòüïðèèlim (1 + α(x)) β(x) ,îò ñòðåìëåíèÿ∞òèïà 1 .α(x)Ëåêöèÿ 8.

Ïðè¼ìû ðàñêðûòèÿ íåîïðåäåë¼ííîñòåé137Ðàñêðûòü íåîïðåäåë¼ííîñòü ýòî çíà÷èò îïðåäåëèòü ïîâåäåíèåâûðàæåíèÿ, ïðèâîäÿùåãî ê äàííîé íåîïðåäåë¼ííîñòè, è íàéòè åãî ïðåäåë.Ðàññìîòðèì íåñêîëüêî ïðèåìîâ ðàñêðûòèÿ íåîïðåäåë¼ííîñòåé ðàçëè÷íîãî òèïà.Ïðèìåð 8.6.3x+5.x→∞ 2x+7Íàéòè limÐ å ø å í è å: äàííîì ïðèìåðå ÷èñëèòåëü è çíàìåíàòåëü áåñêîíå÷íî áîëüøèå âåëè÷èíû, ò.å.

èìååò ìåñòî íåîïðåäåë¼ííîñòü òè∞ïà. ×òîáû ðàñêðûòü ýòó íåîïðåäåë¼ííîñòü, ðàçäåëèì ÷èñëèòåëü è∞çíàìåíàòåëü íà x. Ïîëó÷èìlimx→∞x→∞3x + 53 + 5/x3= lim= ,x→∞2x + 72 + 7/x257èñòðåìèòñÿ ê íóëþ.xxÀíàëîãè÷íî ïîíÿòèþ ýêâèâàëåíòíîñòè áåñêîíå÷íî ìàëûõ ôóíêöèéòàê êàê ïðèêàæäàÿ èç äðîáåéìîæíî ââåñòè ïîíÿòèå ýêâèâàëåíòíîñòè áåñêîíå÷íî áîëüøîé ôóíêöèè.Îïðåäåëåíèå 8.6. Ôóíêöèè N (x) è M (x) áåñêîíå÷íî áîëüøèå ïðèx → a íàçûâàþòñÿ ýêâèâàëåíòíûìè, åñëè ïðåäåë èõ îòíîøåíèÿN (x)lim M= 1.(x)x→aÎ÷åâèäíî, ÷òî ìíîãî÷ëåíâàëåíòåí ïðèx→∞Pn (x) = bn xn + bn−1 xn−1 + · · · + b0÷ëåíó ñ íàèâûñøèì ïîêàçàòåëåì ñòåïåíèýêâèbn xn ,òàê êàêPn (x)bn xn + bn−1 xn−1 + · · · + b0=lim=1x→∞ bn xnx→∞bn xnlimÑëåäîâàòåëüíî, äëÿ îòíîøåíèÿ ìíîãî÷ëåíîâ ïðèx → ∞áóäåòèìåòü ìåñòî ïðàâèëîQm (x)am xm + am−1 xm−1 + ...

+ a0= lim=x→∞ Pn (x)x→∞ bn xn + bn−1 xn−1 + ... + b0lim(8.1)amam xmlim xm−n=x→∞x→∞ bn xnbn= lim•åñëè ñòåïåíü ÷èñëèòåëÿ ìåíüøå ñòåïåíè çíàìåíàòåëÿòî ïðåäåë ðàâåí íóëþ;(m < n),138Ëåêöèÿ 8. Ïðè¼ìû ðàñêðûòèÿ íåîïðåäåë¼ííîñòåé•åñëè ñòåïåíü ÷èñëèòåëÿ áîëüøå ñòåïåíè çíàìåíàòåëÿòî ïðåäåë ðàâåí•+∞èëèåñëè ñòåïåíè ÷èñëèòåëÿ è çíàìåíàòåëÿ ðàâíûïðåäåë ðàâåí êîíå÷íîìó ÷èñëóÏðèìåð 8.7.Ð å ø å í è å:(m > n),−∞;am.bm(m = n),òî8x3 +3x−5.32x→∞ 4x −2x +3Íàéòè limÑîãëàñíî ïðàâèëó (8.1)8x3 + 3x − 58x3=lim= 2.x→∞ 4x3 − 2x2 + 3x→∞ 4x3limÏðèìåð 8.8.Ð å ø å í è å:5x5 −2x2 +3.4x→−∞ 2x +3x−5Íàéòè limÑîãëàñíî ïðàâèëó (8.1)5x − 2x2 + 35x55xlim= lim= lim= −∞.x→−∞ 2x4 + 3x − 5x→−∞ 2x4x→−∞ 25Ïðàâèëî, ïîäîáíîå (8.1), ñïðàâåäëèâî è äëÿ èððàöèîíàëüíûõ âûðàæåíèé.Ïðèìåð 8.9.√√3 3 x2 +5x−4+ x√√63 .4x→∞ 2x −3x+2+2 xÍàéòè lim√3− 4 ∼ 3 x2 = 3x2/3 ,= x1/3 .

Íàèâûñøàÿ√33 x2 + 5x − 4 ∼ 3x2/3 , çíàìå√3Ð å ø å í è å: Òàê êàê ïðè x → ∞ 3 x2 + 5x√√√√√6x = x1/2 6 2x4 − 3x + 2 ∼ 2x4 = 6 2x2/3 , 3 xx â ÷èñëèòåëå èìååò√ ñëàãàåìîå√62x4 − 3x + 2 ∼ 6 2x2/3 , ñëåäîâàòåëüíî√√3 3 x2 + 5x − 4 + x3x2/33√√lim √=lim=.√666x→∞2x4 − 3x + 2 + 2 3 x x→∞ 2x2/32ñòåïåíüíàòåëåÏðè âû÷èñëåíèè ïðåäåëîâ, ñîäåðæàùèõ èððàöèîíàëüíûå âûðàæåíèÿ, òàêæå ÷àñòî èñïîëüçóþò ñëåäóþùèå ïðèåìû:••ââåäåíèå íîâîé ïåðåìåííîé äëÿ ïîëó÷åíèÿ ðàöèîíàëüíîãî âûðàæåíèÿ, åñëè ýòî âîçìîæíî;ïåðåâîä èððàöèîíàëüíîñòè èç çíàìåíàòåëÿ â ÷èñëèòåëü è íàîáîðîò, ïðè ýòîì èñïîëüçóþòñÿ ôîðìóëû òîæäåñòâåííûõ ïðåîáðàçîâàíèé àëãåáðàè÷åñêèõ âûðàæåíèé:√√√√√√( x + y) · ( x − y) = ( x)2 − ( y)2 = x − y(x > 0, y > 0),Ëåêöèÿ 8.

Ïðè¼ìû ðàñêðûòèÿ íåîïðåäåë¼ííîñòåé139√√√√√√√3( 3 x ± 3 y) · ( x2 ∓ 3 x · y + 3 y 2 ) = ( 3 x)3 ± ( 3 y)3 ==x±yÏðèìåð 8.10.Ð å ø å í è å:(x > 0, y > 0).√Íàéòè lim ( x2 − 1 − x).x→∞Çäåñü íåîïðåäåëåííîñòü òèïà∞ − ∞.Óìíîæèìè ðàçäåëèì âûðàæåíèå ïîä ïðåäåëîì íà ñîïðÿæ¼ííîå åìó âûðàæåíèå√x2 − 1 + x.√√√( x2 − 1 − x)( x2 − 1 + x)2√lim ( x − 1 − x) = lim=x→∞x→∞x2 − 1 + xx2 − 1 − x2−1= lim √= lim √= 0.x→∞x2 − 1 + x x→∞ x2 − 1 + xÏðèìåð 8.11.√√Íàéòè lim ( 3 x + 1 − 3 x).Ð å ø å í è å:Îïÿòü ìû èìååì äåëî ñ íåîïðåäåë¼ííîñòüþ òèïà∞ − ∞.x→∞Óñòðàíèòü ýòó íåîïðåäåëåííîñòü ìîæíî, åñëè óìíîæèòü èðàçäåëèòü èñõîäíîå âûðàæåíèå íà íåïîëíûé êâàäðàò ñóììû äâóõ âûðàæåíèé. Ïîñëå ýòîãî ìîæíî ïðèìåíèòü ôîðìóëó ðàçíîñòè êóáîâ äâóõâûðàæåíèé.√√lim ( 3 x + 1 − 3 x) =x→∞√√√√ √√3( 3 x + 1 − 3 x)( 3 (x + 1)2 + 3 x + 1 3 x + x2 )√√= lim=√√33x→∞(x + 1)2 + 3 x + 1 3 x + x2√√( 3 x + 1)3 − ( 3 x)3√ =√= lim √x→∞ 3 (x + 1)2 + 3 (x + 1)x + 3 x2x+1−x√= lim √=√332x→∞(x + 1) + 3 x2 + x + x21√ = 0.= lim √√x→∞ 3 (x + 1)2 + 3 x2 + x + 3 x2Åñëè ìû ïðèìåíèì ê ïðèìåðàì 8.10 è 8.11 ïðàâèëî ýêâèâàëåíòíîñòè áåñêîíå÷íî áîëüøèõ ôóíêöèé, òî ïîëó÷èì òîò æå ðåçóëüòàò.Îäíàêî ñòîèò íåñêîëüêî èçìåíèòü óñëîâèÿ ïðèìåðîâ è ðåçóëüòàò îêàæåòñÿ äðóãèì.140Ëåêöèÿ 8.

Ïðè¼ìû ðàñêðûòèÿ íåîïðåäåë¼ííîñòåéÏðèìåð 8.12.√Íàéòè lim ( x2 + 3x − 1 − x)x→∞Ð å ø å í è å:√√2 + 3x − 1 − x) · ( x2 + 3x − 1 + x)√(x√lim ( x2 + 3x − 1−x) = lim=x→∞x→∞( x2 + 3x − 1 + x)x2 + 3x − 1 − x23x − 1= lim √= lim √=x→∞ ( x2 + 3x − 1 + x)x→∞ ( x2 + 3x − 1 + x)√3x − 13= |Çíàìåíàòåëüx2 + 3x − 1 + x ∼ 2x| = lim= .x→∞2x2Ïîýòîìó ïðèìåíÿòü ïðèíöèï ýêâèâàëåíòíîñòè â ñëó÷àå ðàçíîñòèN (x) − M (x) ìîæíî, åñëè êîýôôèöèíàèâûñøåãî ïîðÿäêà x â N (x) è M (x) íå ðàâíû,áåñêîíå÷íî áîëüøèõ ôóíêöèéåíòû ïðè ñòåïåíÿõïîñêîëüêó â ïðîòèâíîì ñëó÷àå ãëàâíóþ ðîëü ìîãóò èãðàòü ñëàãàåìûåíèçøåãî ïîðÿäêà.×òîáû îïðåäåëèòü ïðåäåë äðîáíî-ðàöèîíàëüíîé ôóíêöèè â ñëó÷àå,êîãäà ïðèx → x0÷èñëèòåëü è çíàìåíàòåëü èìåþò ïðåäåëû, ðàâíûå0), íàäî ÷èñëèòåëü è çíàìåíàòåëü äðîáè ðàç0è ïåðåéòè ê âû÷èñëåíèþ ïðåäåëà. Åñëè æå è ïîñëåíóëþ (íåîïðåäåëåííîñòüäåëèòü íà(x − x0 )ýòîãî ÷èñëèòåëü è çíàìåíàòåëü íîâîé äðîáè èìåþò ïðåäåëû, ðàâíûåíóëþ ïðèx → x0 ,òî íàäî ïðîèçâåñòè ïîâòîðíîå äåëåíèå íà(x − x0 )è ò.ä., äî òåõ ïîð, ïîêà íå áóäåò ïîëó÷åí îêîí÷àòåëüíûé ðåçóëüòàò. íåêîòîðûõ ñëó÷àÿõ ýòó íåîïðåäåëåííîñòü ìîæíî ëåãêî ðàñêðûòü,ðàçëîæèâ ïðåäâàðèòåëüíî ÷èñëèòåëü è çíàìåíàòåëü íà ñîìíîæèòåëèè ñîêðàòèâ íà(x − x0 ).Ïðèìåð 8.13.Ð å ø å í è å:x2 −92 −3x .xx→3Íàéòè limÏðè ïîäñòàíîâêå ïðåäåëüíîãî çíà÷åíèÿ àðãóìåíòàx=3÷èñëèòåëü è çíàìåíàòåëü äðîáè ñòðåìÿòñÿ ê íóëþ.

Èìååò ìåñòî0. Ðàçëîæèì âûðàæåíèå â ÷èñëèòåëå è çíàìåíàòåëå0è ïðîèçâåäåì ñîêðàùåíèå íà (x − 3):íåîïðåäåëåííîñòüx2 − 9(x − 3)(x + 3)x+3lim= lim= lim= 2.x→3 x2 − 3xx→3x→3x(x − 3)xÏðè ðåøåíèè ðÿäà ïðèìåðîâ öåëåñîîáðàçíî èñïîëüçîâàòü ïîíÿòèåýêâèâàëåíòíîñòè áåñêîíå÷íî ìàëûõ ôóíêöèé.Ëåêöèÿ 8. Ïðè¼ìû ðàñêðûòèÿ íåîïðåäåë¼ííîñòåéÏðèìåð 8.14.1415x2 +2 sin 2x+tg2 xarctg 3x+5x2x→0Íàéäèòå limx → 0 sin 2x ∼ 2x, tg x ∼ x, arctg 3x ∼ 3x.x ÷èñëèòåëÿ èìååò ñëàãàåìîå3 sin 2x, çíàìåíàòåëÿ arctg 3x.Ñëåäîâàòåëüíî, ñîãëàñíî òåîðåìå 8.3,22÷èñëèòåëü 5x + 3 sin 2x + tg x ∼ 3 sin 2x ∼ 6x, çíàìåíàòåëü arctg 3x +2+ 5x ∼ arctg 3x ∼ 3x. È íà îñíîâàíèè òåîðåìû 8.15x2 + 2 sin 2x + tg2 x6xlim= lim= 2.2x→0x→0 3xarctg 3x + 5xÐ å ø å í è å: Òàê êàê ïðèÈìååì, ÷òî íèçøèé ïîðÿäîê ìàëîñòè ïîÏðè íàõîæäåíèè ïðåäåëîâ âèäàlim [φ(x)]ψ(x) = C(8.2)x→añëåäóåò èìåòü â âèäó, ÷òî:•åñëè ñóùåñòâóþò êîíå÷íûå ïðåäåëûA = lim φ(x) è B = lim ψ(x),x→ax→aC = AB ;• åñëè lim φ(x) = A ̸= 1òîè lim ψ(x) = ±∞, òî âîïðîñ î íàõîæx→ax→aäåíèè ïðåäåëà ðåøàåòñÿ íåïîñðåäñòâåííî ïîäñòàíîâêîé ïðå-•äåëüíîãî çíà÷åíèÿ àðãóìåíòà;åñëèlim φ(x) = A = 1 è lim ψ(x) = ∞, òî ïîëàãàþò φ(x) = 1+x→aα(x), ãäåx→a+α(x) → 0 ïðè x → a è, ñëåäîâàòåëüíî,1C = lim [φ(x)]ψ(x) = lim {[1 + α(x)] α(x) }α(x)ψ(x) =x→alim α(x)ψ(x)x→alim [φ(x)−1]ψ(x)= e= e1x → a lim [1 + α(x)] α(x) = e.x→ax→ax→aÏðèìåð 8.15.Íàéòè limx→0,( sin 4x )1+x2xòàê êàê â äàííîì ñëó÷àå ïðè.lim sin2x4x = 2 è lim (x + 1) = 1.x→0()1+xsin 4xlim= 21 = 2.x→02xÐ å ø å í è å: Çäåñüx→0Ñëåäîâàòåëüíî,142Ëåêöèÿ 8.

Ïðè¼ìû ðàñêðûòèÿ íåîïðåäåë¼ííîñòåéÏðèìåð 8.16.Íàéòè limx→∞Ð å ø å í è å: Çäåñüïîýòîìó(limx→∞Ïðèìåð 8.17.(x2 +14x2 −3x2 +12x→∞ 4x −3limx2 + 14x2 − 3)x+3Íàéòè limx→∞)x+3= lim.( 1+ 1 )4−x→∞x23x2=1, lim (x + 3)4 x→∞= ∞,111= ( )∞ = ∞ == 0.44∞( x+8 )xx−2.1+ 8lim x+8 = lim 1− x2 = 1, lim x = ∞, ò.å. èìååòx→∞ x−2x→∞x→∞x∞ìåñòî íåîïðåäåëåííîñòü 1 . Ïðîèçâåäÿ óêàçàííûå âûøå ïðåîáðàçîâàíèÿ (φ(x) = 1 + α(x)), ïîëó÷èì()x[()]x[()]xx+8x+810lim= lim 1 +−1= lim 1 +=x→∞x→∞x→∞x−2x−2x−2Ð å ø å í è å: Çäåñü{[= limx→∞101+x−210} x−2·x] x−210lim10x= ex→∞ x−2 = e10 . äàííîì ñëó÷àå, íå ïðèáåãàÿ ê îáùåìó ïðèåìó, ìîæíî íàéòè ïðåäåë ïðîùå:()x()8 xx)2 xx[(8x) x8 ]8lim 1 +10[({ −2 })− x2 ]−2 = e .x→∞x→∞ 1 −lim 1 + xx→∞()k x= ek , èëè â áîëåå äàëüíåéøåì ïîëåçíî ïîìíèòü, ÷òî lim 1 +xlimx+8x−2= lim (1+=x→∞x→∞îáùåì âèäå) xam bxn(m(a )bxna ) xalim abxn−mlim 1 + m= ex→∞== lim1+ mx→∞x→∞xx(∞, n > m; ab > 00,n > m, ab < 0=abe,n = m; 1,n < m.(8.3)Ïðàêòè÷åñêîå çàíÿòèå 8.

Òåîðèÿ ïðåäåëîâ (ïðîäîëæåíèå)143Ïðàêòè÷åñêîå çàíÿòèå 8.Òåîðèÿ ïðåäåëîâ (ïðîäîëæåíèå)Ñðàâíèòü áåñêîíå÷íî ìàëûå âåëè÷èíû α(t) = t · sin2 tè β(t) = 2t · sin t ïðè t → 0.Ïðèìåð 8.1.Ð å ø å í è å:Íàéäåì îòíîøåíèå áåñêîíå÷íî ìàëûõ ôóíêöèéαt · sin2 t11= lim= lim sin t = · 0 = 0,t→0 βt→0 2t · sin t2 t→02limαò.å.åñòü áåñêîíå÷íî ìàëàÿ áîëåå âûñîêîãî ïîðÿäêà ÷åìÏðèìåð 8.2.β.Íàéòè lim (5x + 2).x→4Ð å ø å í è å:  ïðîñòåéøèõ ñëó÷àÿõ íàõîæäåíèå ïðåäåëà ñâîäèòñÿê ïîäñòàíîâêå â âûðàæåíèå ïðåäåëüíîãî çíà÷åíèÿ àðãóìåíòà.

Òàê êàêïðèx → 4 −→ 5 + 2 = 22,òî â ðåçóëüòàòålim (5x + 2) = 22.x→4(Ïðè îòûñêàíèè ïðåäåëà îòíîøåíèÿ äâóõ öåëûõ ìíîãî÷ëåíîâP (x)Q(x))∞, îáà ÷ëåíà∞nîòíîøåíèÿ ðåêîìåíäóåòñÿ ïðåäâàðèòåëüíî ðàçäåëèòü íà x , ãäå n ïðè→ ∞, ò.å. êîãäà èìååò ìåñòî íåîïðåäåëåííîñòü âèäàíàèâûñøàÿ ñòåïåíü ýòèõ ìíîãî÷ëåíîâ, èëè âîñïîëüçîâàòüñÿ ïðàâèëîì(8.1).Ïðèìåð 8.3.Ð å ø å í è å:5x−4.x→∞ 4x+2Íàéòè limÏðèx→∞÷èñëèòåëü è çíàìåíàòåëü èññëåäóåìîéäðîáè íåîãðàíè÷åííî âîçðàñòàþò. ýòîì ñëó÷àå ãîâîðÿò, ÷òî èìååò ìå∞ñòî íåîïðåäåëåííîñòü âèäà. Ðàçäåëèâ íà x îäíîâðåìåííî ÷èñëèòåëü∞è çíàìåíàòåëü äðîáè, ïîëó÷èì:5x − 4lim= limx→∞ 4x + 2x→∞x→∞5x−4x4x+2x5−= limx→∞ 4 +4x2x5= ,442êàæäàÿ èç äðîáåéèñòðåìèòñÿ ê íóëþ.xx3x−45x5Ïî ïðàâèëó (8.1) lim= lim 4x = 4 .x→∞ 4x+2x→∞òàê êàê ïðè144Ïðàêòè÷åñêîå çàíÿòèå 8.