Теория пределов и числовые ряды (Конспект), страница 7

Òåîðèÿ ïðåäåëîâ (ïðîäîëæåíèå)Ïðèìåð 8.4.3x2 +2x+1.2x→∞ 2x +3Íàéòè limÐ å ø å í è å: Ïðè x → ∞ ìû èìååì äåëî ñ íåîïðåäåë¼ííîñòüþ∞2. Ðàçäåëèì íà x îäíîâðåìåííî ÷èñëèòåëü è çíàìåíàòåëü äðîáè,∞ïîëó÷èìòèïà3x2 + 2x + 1lim= limx→∞x→∞2x2 + 3òàê êàê ïðèx→∞Ïî ïðàâèëó (8.1)Ïðèìåð 8.5.3x2 +2x+1x22x2 +3x23 + x2 + x123= ,3x→∞22 + x2= lim2 1, , 3 → 0.x x2 x22lim 3x2 = 32 .x→∞ 2xêàæäàÿ èç äðîáåélimx→∞3x2 +2x+12x2 +3=x2 −2x.x→−∞ 6x+7Íàéòè limÐ å ø å í è å: Ïîñêîëüêó ïðè x → −∞ îïÿòü èìååì äåëî ñ íåîïðåäå∞, ïðè èññëåäîâàíèè ïðåäåëà îòíîøåíèÿ äâóõ ìíîãî÷ëåíîâ∞âîñïîëüçóåìñÿ ïðåäëîæåííîé âûøå ðåêîìåíäàöèåé. Ñòàðøàÿ ñòåïåíüë¼ííîñòüþðàññìàòðèâàåìûõ ìíîãî÷ëåíîâ ðàâíà 2, ïîýòîìó ðàçäåëèì ÷èñëèòåëü2è çíàìåíàòåëü äðîáè íà x .x2 − 2xlim= limx→−∞ 6x + 7x→−∞x2 −2xx26x+7x2= limx→−∞1 − x21= −∞.67 =−0+ x2xÏî ïðàâèëó (8.1)x2 − 2xx26= lim= lim= −∞x→−∞ 6x + 7x→−∞ 6xx→−∞ xlimÏðèìåð 8.6.25x√ −3x+4 .4x4 +5x→∞Íàéòè lim∞.

 ñîîò∞nâåòñòâèè ñ ðåêîìåíäàöèåé ðàçäåëèì ÷èñëèòåëü è çíàìåíàòåëü íà x ,Ð å ø å í è å:  äàííîì ñëó÷àå èìååì íåîïðåäåëåííîñòüãäå n ñòàðøàÿ ñòåïåíü ìíîãî÷ëåíîâ. Ó÷èòûâàÿ, ÷òî â çíàìåíàòåëåx4 ñòîèò ïîä êâàäðàòíûì êîðíåì, äåëèì âñå íà x2 .5x2 − 3x + 4lim √= limx→∞x→∞4x4 + 55x2 −3x+4x2√4x4 +5x25 − x3 + x42= lim √=x→∞4 + x545−0+05= √= .24+0Ïðàêòè÷åñêîå çàíÿòèå 8. Òåîðèÿ ïðåäåëîâ (ïðîäîëæåíèå)145Äëÿ ðåøåíèÿ ýòîãî ïðèìåðà ìîæíî âîñïîëüçîâàòüñÿ ïðàâèëîì ýêâè√√âàëåíòíîñòè áåñêîíå÷íî áîëüøèõ âåëè÷èí4x4 + 5 ∼ 4x4 = 2x2 è5x2 − 3x + 45x2 − 3x + 45x25√= lim=lim= .22x→∞x→∞x→∞ 2x2x24x4 + 5limlim P (x) , ãäåx→a Q(x)îäèí èç íèõ â òî÷êå aP (x) è Q(x) öåëûå ìíîãî÷ëåíû è õîòÿ áû̸= 0 íàõîäèòñÿ íåïîñðåäñòâåííîé ïîäñòàíîâêîé â ôóíêöèþ ïðåäåëüíîãî çíà÷åíèÿ àðãóìåíòà x = a.

Åñëè æåP (x)P (a) = Q(a) = 0 è èìååò ìåñòî íåîïðåäåëåííîñòü 00 , òî äðîáü Q(x)ðåêîìåíäóåòñÿ ñîêðàòèòü îäèí èëè íåñêîëüêî ðàç íà ðàçíîñòü (x − a).ÏðåäåëÏðèìåð 8.7.x3 +12 +1 .xx→−1Íàéòè lim3Ð å ø å í è å: Ïðè ïîäñòàíîâêå x = −1 â ÷èñëèòåëü èìååì x +1 == −13 + 1 = −1 + 1 = 0. Ïðè ïîäñòàíîâêå x = −1 â çíàìåíàòåëü x2 + 1 = −12 + 1 = 1 + 1 = 2. Ñëåäîâàòåëüíî,x3 + 10= = 0.2x→−1 x + 12limÏðèìåð 8.8.x2 −4.2x→2 x −3x+2Íàéòè limÐ å ø å í è å: Ïðè ïîäñòàíîâêå x = 2 èìååì íåîïðåäåëåííîñòü0. Ðàçëîæèì ÷èñëèòåëü è çíàìåíàòåëü íà ñîìíîæèòåëè è ïðîèç0âåäåì íåîáõîäèìûå ñîêðàùåíèÿ.  ðåçóëüòàòå ïîëó÷èì:âèäàx2 − 4(x − 2)(x + 2)x+2=lim=lim= 4.x→2 x2 − 3x + 2x→2 (x − 2)(x − 1)x→2 x − 1limÂûðàæåíèÿ, ñîäåðæàùèå èððàöèîíàëüíîñòü, ïðèâîäÿòñÿ ê ðàöèîíàëüíîìó âèäó âî ìíîãèõ ñëó÷àÿõ ïóò¼ì ââåäåíèÿ íîâîé ïåðåìåííîé.√1+x−1Ïðèìåð 8.9.

Íàéòè lim √.3x→0 1+x−1Ð å ø å í è å: Ïðè ïîäñòàíîâêå x = 0 èìååì íåîïðåäåëåííîñòü06âèäà . Îáîçíà÷èì 1 + x = y äëÿ òîãî, ÷òîáû ïðè èçâëå÷åíèè êâàä0ðàòíîãî è êóáè÷åñêîãî êîðíåé ïîëó÷èòü öåëûå ñòåïåíè. Ó÷èòûâàÿ, ÷òîïðèx → 0, y → 1, èìååì:√√y6 − 11+x−1y3 − 1√.lim √=lim=limx→0 3 1 + x − 1y→1 3 y 6 − 1y→1 y 2 − 1146Ïðàêòè÷åñêîå çàíÿòèå 8.

Òåîðèÿ ïðåäåëîâ (ïðîäîëæåíèå)Ïðèåì íàõîæäåíèÿ ïîñëåäíåãî ïðåäåëà àíàëîãè÷åí òîìó, êîòîðûé ìûèñïîëüçîâàëè ïðè ðåøåíèè ïðèìåðà 8.8. Äëÿ ðàñêðûòèÿ íåîïðåäåë¼í0íîñòè(ïðè y = 1) ðàçëîæèì íà ìíîæèòåëè ÷èñëèòåëü è çíàìåíàòåëü,0ïðîèçâåäåì íåîáõîäèìûå ñîêðàùåíèÿ è â ðåçóëüòàòå ïîëó÷èìy3 − 1(y − 1)(y 2 + y + 1)y2 + y + 1=lim=lim=y→1 y 2 − 1y→1y→1(y − 1)(y + 1)y+11+1+13== .1+12limÄðóãèì ïðèåìîì íàõîæäåíèÿ ïðåäåëà îò èððàöèîíàëüíîãî âûðàæåíèÿ ÿâëÿåòñÿ ïåðåâîä èððàöèîíàëüíîñòè èç ÷èñëèòåëÿ â çíàìåíàòåëü èëè, íàîáîðîò, èç çíàìåíàòåëÿ â ÷èñëèòåëü. Ïðè ýòîì èñïîëüçóþòñÿ ôîðìóëû òîæäåñòâåííûõ ïðåîáðàçîâàíèé àëãåáðàè÷åñêèõ âûðàæåíèé:√√√ √√√( x + y)( x − y) = ( x)2 − ( y)2 = x − y(x > 0, y > 0)√√√√√√√3( x ± y)( x2 ∓ 3 xy + 3 y 2 ) = ( 3 x)3 ± ( 3 y)3 = x ± y (x > 0, y > 0).√Ïðèìåð 8.10.

Íàéòè lim ( x2 − 5x + 6 − x).x→∞Ð å ø å í è å: Ïðè→ ∞ ìû èìååì íåîïðåäåëåííîñòü âèäà ∞ − ∞.Óìíîæèì è ðàçäåëèì âûðàæåíèå, ñòîÿùåå ïîä ïðåäåëîì, íà âûðàæåíèå, åìó ñîïðÿæ¼ííîå (íà ñóììó òàêèõ æå ñëàãàåìûõ).  äàííîì√ñëó÷àå íà ( x2 − 5x + 6+x). Ïîñëå ýëåìåíòàðíûõ ïðåîáðàçîâàíèé ïîëó÷èì:√lim ( x2 − 5x + 6 − x) =√√ x→∞( x2 − 5x + 6 − x)( x2 − 5x + 6 + x)√== limx→∞x2 − 5x + 6 + xx2 − 5x + 6 − x26 − 5x√= lim= lim √.22x→∞x→∞x − 5x + 6 + xx − 5x + 6 + xÎ÷åâèäíî, ÷òî ïðè x → ∞ ïîñëåäíèé ïðåäåë ïðèâîäèòñÿ êíåîïðåäå∞ë¼ííîñòè âèäà. Ðàçäåëèì ÷èñëèòåëü è çíàìåíàòåëü îäíîâðåìåííî∞níà x , ãäå n ñòàðøàÿ ñòåïåíü ìíîãî÷ëåíîâ.

 äàííîì ñëó÷àå íà x.6 − 5xlim √= lim2x→∞x − 5x + 6 + x x→∞√6−5xxx2 −5x+6+xx6−5x= lim √=x→∞1 − x5 + x62 + 10−5−5−55=√=√==− .1+121−0+0+11+1Ïðàêòè÷åñêîå çàíÿòèå 8. Òåîðèÿ ïðåäåëîâ (ïðîäîëæåíèå)Ïðèìåð 8.11.Íàéòè lim (x −x→∞Ð å ø å í è å:√3x3 + 8x2 ). äàííîì ïðèìåðå ïðè∞ − ∞.äåëåííîñòü âèäà147x → ∞èìååì íåîïðå-Óìíîæèì è ðàçäåëèì âûðàæåíèå, ñòîÿùååïîä çíàêîì ïðåäåëà, íà íåïîëíûé êâàäðàò ñóììû ñëàãàåìûõ, ÷òîáû âèòîãå ïîëó÷èòü â ÷èñëèòåëå ôîðìóëó ðàçíîñòè êóáîâ äâóõ ÷èñåë:lim (x −x→∞= lim(x −x→∞√3x3 + 8x2 ) =√√√3x3 + 8x2 )(x2 + x 3 x3 + 8x2 + 3 (x3 + 8x2 )2 )√√=x2 + x 3 x3 + 8x2 + 3 (x3 + 8x2 )2√x3 − ( 3 x3 + 8x2 )3√√= lim=x→∞ x2 + x 3 x3 + 8x2 + 3 (x3 + 8x2 )2x3 − (x3 + 8x2 )√√=x→∞ x2 + x 3 x3 + 8x2 + 3 (x3 + 8x2 )2= lim−8x2√√=x→∞ x2 + x 3 x3 + 8x2 + 3 (x3 + 8x2 )2= lim√x→∞ 2x (1 + 3 1 += −8 lim√x→∞1+ 3 1+= −8 limÏðèìåð 8.12.18x+√3x28x√+(1 + x8 )23=(1 + x8 )2 )= −8 ·18=− .1+1+13√3−√5+x.x→4 1− 5−xÍàéòè limÐ å ø å í è å:Ïðè x = 4 èìååì äåëî ñ íåîïðåäåë¼ííîñòüþ0âèäà .

Ðàñêðîåì ýòó íåîïðåäåëåííîñòü ñëåäóþùèì îáðàçîì: ÷èñëè0òåëü óìíîæèì è ðàçäåëèì íà âûðàæåíèå, ñîïðÿæ¼ííîå ÷èñëèòåëþ→ (3 +√ìåíàòåëþ5 + x), a√çíàìåíàòåëü→ (1 + 5 − x). íà âûðàæåíèå, ñîïðÿæ¼ííîå çíà-148Ïðàêòè÷åñêîå çàíÿòèå 8. Òåîðèÿ ïðåäåëîâ (ïðîäîëæåíèå)√√√√5+x(3 − 5 + x) · 3+3− 5+x3+ 5+x√√lim= lim=√x→4 1 −5 − x x→4 (1 − 5 − x) · 1+√5−x1+ 5−x√√√(3 − 5 + x)(3 + 5 + x)(1 + 5 − x)√√√= lim=x→4 (1 −5 − x)(1 + 5 − x)(3 + 5 + x)√√(9 − ( 5 + x)2 )(1 + 5 − x)√√== limx→4 (1 − ( 5 − x)2 )(3 +5 + x)√(9 − (5 + x))(1 + 5 − x)√= lim=x→4 (1 − (5 − x))(3 +5 + x)√√(9 − 5 − x))(1 + 5 − x)(4 − x))(1 + 5 − x)√√= lim= lim=x→4 (1 − 5 + x))(3 +5 + x) x→4 (−4 + x))(3 + 5 + x)√1+ 5−x1+111√= −1 · lim= −1 ·= −1 · = − .x→4 3 +3+3335+xÏðè íàõîæäåíèè ïðåäåëîâ îò ðàçíîñòè äâóõ äðîáåé, êîãäà èìååò∞ − ∞,ìåñòî íåîïðåäåëåííîñòü âèäàðåêîìåíäóåòñÿ ïðåäâàðèòåëüíîïðèâåñòè äðîáè ê îáùåìó çíàìåíàòåëþ.Ïðèìåð 8.13.Ð å ø å í è å:∞ − ∞.Íàéòè lim(x→3Ïðèx=31x−3−6x2 −9).èìååì äåëî ñ íåîïðåäåë¼ííîñòüþ âèäàÂîñïîëüçóåìñÿ ðåêîìåíäàöèåé è ïðèâåäåì äðîáü ê îáùåìóçíàìåíàòåëþ.

Ïîñëå ýëåìåíòàðíûõ ïðåîáðàçîâàíèé ïîëó÷èì:()16x+3−6lim− 2= lim=x→3x→3 (x − 3)(x + 3)x−3 x −9x−3111= lim= lim== .x→3 (x − 3)(x + 3)x→3 x + 33+36Ðàññìîòðèì åù¼ ðàç íåêîòîðûå ïðèìåðû ïðåäûäóùåãî ïðàêòè÷åñêîãî çàíÿòèÿ è èñïîëüçóåì äëÿ èõ ðåøåíèÿ ïîíÿòèå ýêâèâàëåíòíîñòèáåñêîíå÷íî ìàëûõ âåëè÷èí è òåîðåìó 8.1.Ïðèìåð 7.8.Ð å ø å í è å:Íàéòè limx→0sin 5x.xsin 5x ∼ 5x ïðè x → 05xsin 5x= lim=5limx→0 xx→0xÒàê êàêÏðàêòè÷åñêîå çàíÿòèå 8. Òåîðèÿ ïðåäåëîâ (ïðîäîëæåíèå)Ïðèìåð 7.11.Íàéòè limÐ å ø å í è å:Òàê êàêÏðèìåð 7.12.Íàéòè lim149x.x→0 tg 5xtg 5x ∼ 5x ïðè x → 0xx1lim= lim=x→0 tg 5xx→0 5x5x→01−cos 5x.x2Ð å ø å í è å:( )225x25x25x25x=2·1 − cos 5x = 2 sin∼2=2242x→025 2x1 − cos 5x252lim=lim=.x→0x→0 x2x222ïðèÑàìîñòîÿòåëüíàÿ ðàáîòàÑðàâíèòü áåñêîíå÷íî ìàëûå âåëè÷èíû:Ïðèìåð 8.14.α = t2 tg t è β = t2 sin2 t ïðè t → 0.Ïðèìåð 8.15.α = 5t2 + 2t5 è β = 3t2 + 2t3 ïðè t → 0.Íàéòè ïðåäåëû ôóíêöèé:x2 −5x+6.2x→3 x −9Ïðèìåð 8.16.x3 +1.2x→−1 x +1Ïðèìåð 8.23.Ïðèìåð 8.17.x2 +2x+5.2x→1 x +1Ïðèìåð 8.24.Ïðèìåð 8.18.4x3 −2x2 +1.3x3 −5x→∞x2 −7x+10.2x→2 x −8x+12√3 x−1Ïðèìåð 8.25.

lim √.4x→1 x−1Ïðèìåð 8.19.x2 +x−1.x→∞ 2x+5Ïðèìåð 8.26.Ïðèìåð 8.20.x3 −1.x→1 x−1Ïðèìåð 8.27.Ïðèìåð 8.21.x2 +3x−10.2x→2 3x −5x−2Ïðèìåð 8.28.Ïðèìåð 8.22.3x2 −2x−1.x3 +4x→∞Ïðèìåð 8.29.limlimlimlimlimlimlimlimlim√5(1+x)3 −1.xlimx→0√limx→01+x−1.x√2x+1−3√ .√x→4 x−2− 2limlimx→0x√.31+x−1150Ïðàêòè÷åñêîå çàíÿòèå 8.

Òåîðèÿ ïðåäåëîâ (ïðîäîëæåíèå)Ïðèìåð 8.30.√√1+x− 1−x.xx→0Ïðèìåð 8.31.limÏðèìåð 8.32.limx→∞(√3x+1−)√3x .limx→1(11−x−31−x3).Ðåøèòå ïðèìåðû èç ïðåäûäóùåãî ïðàêòè÷åñêîãî çàíÿòèÿ 7.33, 7.34,7.36, 7.37, 7.38 ñ èñïîëüçîâàíèåì ïîíÿòèÿ ýêâèâàëåíòíîñòè áåñêîíå÷íîìàëûõ âåëè÷èí.Ïðàêòè÷åñêîå çàíÿòèå 9. Íåïðåðûâíîñòü ôóíêöèè151Ïðàêòè÷åñêîå çàíÿòèå 8.

Íåïðåðûâíîñòü ôóíêöèèÏðèìåð 8.1.Èññëåäîâàòü íà íåïðåðûâíîñòü ôóíêöèþy=Ð å ø å í è å: òî÷êå1 + x3.1+xx = −1ôóíêöèÿ íå îïðåäåëåíà, òàê êàê,0.  äðóãèõ òî÷0êàõ äðîáü ìîæíî ñîêðàòèòü íà (1 + x), òàê êàê â íèõ 1 + x ̸= 0. Ëåãêîâûïîëíèâ ïîäñòàíîâêó, ïîëó÷àåì íåîïðåäåëåííîñòüâèäåòü, ÷òî îäíîñòîðîííèå ïðåäåëû ñëåâà è ñïðàâà â òî÷êåx = −1ðàâíû ìåæäó ñîáîé è èõ ìîæíî âû÷èñëèòü:lim y =x→−1−0lim y =x→−1+01 + x3=x→−1+0 1 + xlim(1 + x)(1 − x + x2 )= lim= lim (1 − x + x2 ) = 1 + 1 + 1 = 3.x→−1+0x→−1+01+xÒàêèì îáðàçîì, ïðè x = −1 äàííàÿ ôóíêöèÿ èìååò óñòðàíèìûé ðàç1+x3ðûâ.

Îí áóäåò óñòðàíåí, åñëè ïîëîæèòü, ÷òî ïðè x = −1 ⇒ y == 3.1+xÏðèìåð 8.2.Èññëåäîâàòü íà íåïðåðûâíîñòü ôóíêöèþy = arctg1x−4(ðèñ. 88).yπ/204x−π/2Ðèñ. 88.Ãðàôèê ôóíêöèè1y = arctg x−4152Ïðàêòè÷åñêîå çàíÿòèå 9. Íåïðåðûâíîñòü ôóíêöèèÐ å ø å í è å: Âû÷èñëèì îäíîñòîðîííèå ïðåäåëû ôóíêöèè â òî÷êåx = 4.å¼ ðàçðûâàÏðåäåë ñëåâà 1lim (arctg x−4) = arctg(−∞) = − π2 .x→4−01lim (arctg x−4) = arctg(+∞) = π2 .x→4+0Ïðåäåëû ñëåâà è ñïðàâà ñóùåñòâóþò, íî íå ðàâíû, ñëåäîâàòåëüíî,Ïðåäåë ñïðàâà òî÷êàx=4äëÿ äàííîé ôóíêöèè òî÷êà ðàçðûâà I ðîäà.Ïðèìåð 8.3.x = 2.Ïîêàçàòü, ÷òî ôóíêöèÿ y = 4x2 íåïðåðûâíà â òî÷êåÐ å ø å í è å:Äëÿ ýòîãî íåîáõîäèìî ïîêàçàòü, ÷òî â òî÷êåx=2âûïîëíÿåòñÿ âñå òðè óñëîâèÿ íåïðåðûâíîñòè ôóíêöèè:y = 4x2 îïðåäåëåíà â òî÷êå x = 2 ⇒ f (2) = 16;2ñóùåñòâóåò lim f (x) = lim 4x = 16;1) ôóíêöèÿ2)x→2x→23) ýòîò ïðåäåë ðàâåí çíà÷åíèþ ôóíêöèè â òî÷êåx=2lim f (x) = f (2) = 16.x→2Ïðèìåð 8.4.