Учебно-методическое пособие (1017796), страница 5
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Ýëåìåíòàðíûå ìåòîäû âû÷èñëåíèÿ ïðåäåëàÏðèìåð 1.1. Âû÷èñëåíèå ïðåäåëà ôóíêöèè ïîäñòàíîâêîé (åñëèñîîòâåòñòâóþùàÿ ïîäñòàíîâêà íå ïðèâîäèò ê íåîïðåäåëåííîñòè):5x + 1 5 · 2 + 1 11== .x→2 3x + 23·2+28limÊàÌôåäÃÒðÓàÌ ÂÌÈ-2ÐÝÀ( )0Ðàñêðûòèå íåîïðåäåëåííîñòåé âèäà0x2 − 5x + 6Ïðèìåð 1.2. Âû÷èñëèòü ïðåäåë ôóíêöèè: lim.x→2x2 − 2xÐåøåíèå: Íåïîñðåäñòâåííàÿ ïîäñòàíîâêà â ÷èñëèòåëü è çíàìåíàòåëü ïðåäåëüíîãî çíà÷åíèÿ àðãóìåíòà x = 2 îáðàùàåòèõ â 0 è( )0ïðèâîäèò ê íåîïðåäåëåííîìó âûðàæåíèþ âèäà.0Ðàçëîæèì ÷èñëèòåëü è çíàìåíàòåëü íà ìíîæèòåëè (ïðè ýòîìâûäåëÿåòñÿ ìíîæèòåëü (x − 2)):x2 − 5x + 6(x − 2)(x − 3)x−3lim=lim=limx→2x→2x→2x2 − 2x(x − 2)xx ðåçóëüòàòå íåïîñðåäñòâåííîé ïîäñòàíîâêè â ïîëó÷åííîå âûðàæåíèå ïðåäåëüíîãî çíà÷åíèÿ àðãóìåíòà ïîëó÷àåì:x2 − 5x + 6x−31lim=lim=−x→2x→2x2 − 2xx2√x2 + 9 − 5Ïðèìåð 1.3.
Âû÷èñëèòü lim.x→4x−4( )0Ðåøåíèå: Èìååì íåîïðåäåëåííîñòü âèäà. Äëÿ ðàñêðûòèÿ0ýòîé íåîïðåäåëåííîñòè óìíîæèì ÷èñëèòåëü è çíàìåíàòåëü íà âûðàæåíèå, ñîïðÿæåííîå ÷èñëèòåëþ:(√) (√)√2+9−52+9+52xxx +9−5(√)lim= lim=x→4x→4x−4(x − 4) x2 + 9 + 563ÊàÌôåäÃÒðÓàÌ ÂÌÈ-2ÐÝÀx2 − 16x+44+4 4(√) = lim √= lim==x→4 (x − 4)x→4x2 + 9 + 5x2 + 9 + 5 5 + 5 5√31 + 3x2 − 1.Ïðèìåð 1.4. Âû÷èñëèòü limx→0x2( )0Ðåøåíèå:  äàííîì ïðèìåðå èìååì íåîïðåäåëåííîñòü.0Óìíîæèì ÷èñëèòåëüíà íåïîëíûé êâàäðàò ñóììû,√(√)è çíàìåíàòåëü3322òî åñòü íà1 + 3x + 1 + 3x + 1.  ðåçóëüòàòå ïîëó÷èì√31 + 3x2 − 11 + 3x2 − 1) √]=lim= lim [(√3x→0x→0 x2x21 + 3x2 + 3 1 + 3x2 + 13√)=1= lim (√x→0 3 1 + 3x2 + 3 1 + 3x2 + 1Ðàñêðûòèå íåîïðåäåëåííîñòåé âèäà(∞)∞x2 − 5x + 1Ïðèìåð 1.5.
Âû÷èñëèòü ïðåäåë ôóíêöèè: lim.x→∞3x2 + 7Ðåøåíèå: Äåëèì ÷èñëèòåëü è çíàìåíàòåëü íà íàèáîëüøóþ ñòåïåíü xx2 5x1−+2x − 5x + 1x2 x2 x2 = 1 − 0 + 0 = 1lim=limx→∞x→∞3x2 + 73+033x27+x2x2Çäåñü èñïîëüçóþòñÿ òåîðåìû îá àðèôìåòè÷åñêèõ ñâîéñòâàõïðåäåëà è òåîðåìà 7, èç êîòîðîé ñëåäóåò, ÷òî175= 0, lim 2 = 0, lim 2 = 0.x→∞ xx→∞ xx→∞ xlim64Ðàñêðûòèå íåîïðåäåëåííîñòåé âèäà (∞ − ∞)Ïðèìåð 1.6. Âû÷èñëèòü lim(√x→+∞x2)+ 3x + 1 − x .Ðåøåíèå: Èìååì íåîïðåäåëåííîñòü âèäà (∞ − ∞).
Óìíîæèì èïîäåëèì íà ñîïðÿæåííîå âûðàæåíèålim(√x→+∞x2)+ 3x + 1 − x =) (√)x2 + 3x + 1 − xx2 + 3x + 1 + x)(√= limx→+∞x2 + 3x + 1 + x3x + 1= lim √= [äåëèìx→+∞x2 + 3x + 1 + x÷èñëèòåëü è çíàìåíàòåëü íà ñòàðøóþ ñòåïåíü çíàìåíàòåëÿ]=ÊàÌôåäÃÒðÓàÌ ÂÌÈ-2ÐÝÀ(√3+= lim √x→+∞1+1x13+ 2 +1x x=321.4. Ïåðâûé è âòîðîé çàìå÷àòåëüíûå ïðåäåëûÏåðâûé çàìå÷àòåëüíûé ïðåäåë .sin x=1x→0 xlimsin 5xsin 5x 5x5 5= lim·=1· =x→0 3xx→0 5x3x3 3sin αxsin αxÏðèìåð 1.8. lim= lim· cos βx =x→0 tg βxx→0 sin βxsin αxβxαx1 αα= lim··· cos βx = · · 1 =x→0 αxsin βx βx1 ββÏðèìåð 1.7.
limÂòîðîé çàìå÷àòåëüíûé ïðåäåë (íåîïðåäåëåííîñòü âèäà 1∞ ) .lim (1 + x)1/x = e, ãäå e ≈ 2.718,x→065èëè, ýêâèâàëåíòíî,()x1lim 1 += e.x→∞xÏðè ðàñêðûòèè íåîïðåäåëåííîñòåé âèäà 1∞ (ò.å. åñëè ïðåäåëîñíîâàíèÿ ðàâåí 1, à ïîêàçàòåëü ñòðåìèòñÿ ê áåñêîíå÷íîñòè) íóæíî çàïèñàòü îñíîâàíèå â âèäå 1 + α(x), à â ïîêàçàòåëå âûäåëèòüìíîæèòåëü1.α(x)5x(()x)55 5Ïðèìåð 1.9. lim 1 += lim 1 + = e5x→∞x→∞xxÊàÌôåäÃÒðÓàÌ ÂÌÈ-2ÐÝÀ(()xx+1Ïðèìåð 1.10.
lim= lim 1 +−1 =x→∞x→∞x−12x()x()x − 1 x − 1222 = e2 .= lim 1 += lim 1 +x→∞x→∞x−1x−1x+1x−1)xÇàìå÷àíèå. Çäåñü èñïîëüçóåòñÿ òåîðåìà 8:[] lim g(x)g(x)lim f (x)= lim f (x) x→ax→ax→aÏðèìåð 1.11. lim (1 − 2x)1/xx→0()−1/(2x) −2= lim (1 − 2x)x→0= e−21.5. Áåñêîíå÷íî ìàëûå ôóíêöèèÎïðåäåëåíèå 1.4 Ôóíêöèÿ α(x) íàçûâàåòñÿ áåñêîíå÷íî ìàëîéïðè x → a, åñëè lim α(x) = 0.x→aÏðèìåð 1.12.
α(x) = sin x áåñêîíå÷íî ìàëàÿ ïðè x → 0.66Ïðèìåð 1.13. α(x) = (x − 5)2 áåñêîíå÷íî ìàëàÿ ïðè x → 5.1 áåñêîíå÷íî ìàëàÿ ïðè x → +∞.xÏðèìåð 1.14. α(x) = √Îïðåäåëåíèå 1.5 Ôóíêöèÿ f (x) íàçûâàåòñÿ îãðàíè÷åííîé âÊàÌôåäÃÒðÓàÌ ÂÌÈ-2ÐÝÀíåêîòîðîé ïðîêîëîòîé îêðåñòíîñòè òî÷êè a, åñëè ∃M > 0 òàêîå, ÷òî ïðè íåêîòîðîì δ > 0 äëÿ âñåõ x ∈ (a − δ, a) ∪ (a, a + δ),ò.å.
óäîâëåòâîðÿþùèõ íåðàâåíñòâó 0 < |x−a| < δ , âûïîëíÿåòñÿíåðàâåíñòâî |f (x)| < M .Îñíîâíûå ñâîéñòâà áåñêîíå÷íî ìàëûõ ôóíêöèé1. Ñóììà (èëè ðàçíîñòü) áåñêîíå÷íî ìàëûõ ôóíêöèé áåñêîíå÷íî ìàëàÿ ôóíêöèÿ.2. Ïðîèçâåäåíèå áåñêîíå÷íî ìàëûõ ôóíêöèé åñòü áåñêîíå÷íîìàëàÿ ôóíêöèÿ.Òåîðåìà 1.11 Åñëè lim f (x) = C ̸= 0 è α(x) áåñêîíå÷íî ìàëàÿx→aôóíêöèÿ ïðè x → a, α(x) ̸= 0 â íåêîòîðîé ïðîêîëîòîé îêðåñòf (x)íîñòè òî÷êè a, òî lim= ∞.x→a α(x)Òåîðåìà 1.12 Åñëè |f (x)| < M , òî åñòü f (x) îãðàíè÷åííàÿôóíêöèÿ, α(x) áåñêîíå÷íî ìàëàÿ ïðè x → a, òî ïðîèçâåäåíèåf (x)α(x) áåñêîíå÷íî ìàëàÿ ôóíêöèÿ ïðè x → a.1.6.
Ýêâèâàëåíòíûå áåñêîíå÷íî ìàëûå ôóíêöèèÎïðåäåëåíèå 1.6 Áåñêîíå÷íî ìàëûå ïðè x → a ôóíêöèè α(x) èβ(x) íàçûâàþòñÿ ýêâèâàëåíòíûìè, åñëèα(x)=1x→a β(x)lim67(ïðåäïîëàãàåòñÿ, ÷òî β(x) ̸= 0 â íåêîòîðîé ïðîêîëîòîé îêðåñòíîñòè òî÷êè a).Äëÿ îáîçíà÷åíèÿ ýêâèâàëåíòíûõ áåñêîíå÷íî ìàëûõ ôóíêöèéèñïîëüçóþò ñëåäóþùóþ ñèìâîëèêó α(x) ∼ β(x) ïðè x → a.Ïðèìåð 1.15. sin x ∼ x ïðè x → 0 (ïåðâûé çàìå÷àòåëüíûé ïðåäåë).ÊàÌôåäÃÒðÓàÌ ÂÌÈ-2ÐÝÀÏðèìåð 1.16. 5x2 + 2x ∼ 2x ïðè x → 0 . Ýòî ñëåäóåò èç òîãî,÷òî()25x + 2x5= lim·x+1x→0x→0 22xlimÏðèìåð 1.17. √=111∼ïðè x → +∞ (ïðîâåðèòü5x25x2 + 3x + 1ñàìîñòîÿòåëüíî).Ïðè âû÷èñëåíèè ïðåäåëîâ ïîëåçíû ñëåäóþùèå òåîðåìû î çàìåíå áåñêîíå÷íî ìàëîé ôóíêöèè íà ýêâèâàëåíòíóþ â ïðîèçâåäåíèè è ÷àñòíîì.Òåîðåìà 1.13 Ïóñòü α(x) ∼ β(x) ïðè x → a è f (x) ïðîèç-âîëüíàÿ ôóíêöèÿ, îïðåäåëåííàÿ â íåêîòîðîé ïðîêîëîòîé îêðåñòíîñòè òî÷êè a.
Òîãäà lim (f (x)α(x)) = lim (f (x)β(x)).x→ax→aÒåîðåìà 1.14 Ïðåäåë îòíîøåíèÿ äâóõ áåñêîíå÷íî ìàëûõ ôóíê-öèé íå èçìåíèòñÿ, åñëè èõ çàìåíèòü íà ýêâèâàëåíòíûå ôóíêöèè.sin 5x5x 5= lim=x→0 sin 3xx→0 3x3Óäîáíî ââåñòè ñëåäóþùåå îïðåäåëåíèå.Ïðèìåð 1.18. limÎïðåäåëåíèå 1.7 Ôóíêöèÿ f (x) íàçûâàåòñÿ áåñêîíå÷íî áîëü-øîé ïðè x → a, åñëè lim f (x) = ∞.x→aÒåîðåìà 1.15 Ôóíêöèÿ f (x) áåñêîíå÷íî áîëüøàÿ ïðè x → a ⇔1 áåñêîíå÷íî ìàëàÿ ôóíêöèÿ ïðè x → a.f (x)68Îñíîâíûå ýêâèâàëåíòíîñòè (ïðè x → 0 )sin x ∼ x2tg x ∼ x3arcsin x ∼ x4arctg x ∼ x5x21 − cos x ∼26ex − 1 ∼ xÊàÌôåäÃÒðÓàÌ ÂÌÈ-2ÐÝÀ17ax − 1 ∼ x ln a9 loga (1 + x) ∼8ln(1 + x) ∼ xx10 (1 + x)a − 1 ∼ axln aÒàáëèöó îñíîâíûõ ýêâèâàëåíòíîñòåé óäîáíî ïåðåïèñàòü â áîëååîáùåì âèäå.Ïóñòü α(x) - áåñêîíå÷íî ìàëàÿ ôóíêöèÿ ïðè x → a. Òîãäàèìåþò ìåñòî ñëåäóþùèå ýêâèâàëåíòíîñòè ïðè x → a.1sin (α(x)) ∼ α(x)2tg (α(x)) ∼ α(x)3arcsin (α(x)) ∼ α(x)4arctg (α(x)) ∼ α(x)6eα(x) − 1 ∼ α(x)8ln(1 + α(x)) ∼ α(x)(α(x))25 1 − cos (α(x)) ∼279bα(x) − 1 ∼ α(x) ln blogb (1 + α(x)) ∼α(x)ln b10 (1 + α(x))b − 1 ∼ bα(x)11ïðè x → ∞xxÏðèìåð 1.20.
tg(x − 2) ∼ x − 2 ïðè x → 2Ïðèìåð 1.19. sin √ ∼ √691.7. Ïðèìåíåíèå òàáëèöû ýêâèâàëåíòíîñòåé êâû÷èñëåíèþ ïðåäåëîâÏðèâåäåì ïðèìåðû èñïîëüçîâàíèÿ òåîðåì î çàìåíå áåñêîíå÷íîìàëîé ôóíêöèè íà ýêâèâàëåíòíóþ â ïðîèçâåäåíèè è ÷àñòíîì ïðèâû÷èñëåíèè ïðåäåëîâ. ñëåäóþùèõ ïðèìåðàõ âû÷èñëèì ïðåäåëû ôóíêöèé, èñïîëüçóÿ òàáëèöó îñíîâíûõ ýêâèâàëåíòíîñòåé.7x7x7Ïðèìåð 1.21.
lim 2x 4 = lim 4 =x→0 e − 1x→0 2x87x 7x 2x(ò.ê. arctg∼ , e − 1 ∼ 2x ïðè x → 0).44sin2 3x(3x)29Ïðèìåð 1.22. lim 2= lim=x→0 ln (1 + 2x)x→0 (2x)24(ò.ê. sin 3x ∼ 3x, ln(1 + 2x) ∼ 2x ïðè x → 0).ÊàÌôåäÃÒðÓàÌ ÂÌÈ-2ÐÝÀarctg5tg 3x − 1tg(3x) · ln 5Ïðèìåð 1.23. lim= lim=x→0 arcsin xx→0x3x · ln 5= lim= 3 ln 5x→0x(2x)21 − cos 2x−2x222Ïðèìåð 1.24. lim=lim=lim=−x→0 ln (1 − 3x2 )x→0 −3x2x→0 3x23eαx − eβxÏðèìåð 1.25. lim=x→0 sin αx − sin βx( (α−β)x)βxe e−11 · (α − β)x= lim= lim=1x→0(α − β)x(α + β)x x→0(α − β)x2 sin· cos2··1222ln[1 − (1 − cos x)]ln(cos x)Ïðèìåð 1.26.
lim=lim=x→0x→0 (1 + x2 )3 − 1(1 + x2 )3 − 1()( x )22 x2 xln 1 − 2 sin−2−2 sin222 = −1= lim=lim=limx→0 (1 + x2 )3 − 1x→0x→03x23x2670cos 4x − cos 2x−2 sin 3x · sin x=lim=x→0x→0arctg2 xarctg2 xÏðèìåð 1.27. lim−2 · 3x · x= −6x→0x2√√331x3 + 7x4x3= lim= , òàê êàê x3 + 7x4 ∼Ïðèìåð 1.28. limx→0 ln(1 + 5x)x→0 5x53x , ln(1 + 5x) ∼ 5x ïðè x → 0.= limÊàÌôåäÃÒðÓàÌ ÂÌÈ-2ÐÝÀ12x3 − 7x6 + sin 5x5xÏðèìåð 1.29. lim=lim=.
Çäåñü èñx→0x→0 10x10x − 2x42ïîëüçóåì òî, ÷òî èìåþò ìåñòî ýêâèâàëåíòíîñòè 2x3 −7x6 +sin 5x ∼5x , 10x − 2x4 ∼ 10x ïðè x → 0 (ïðîâåðèòü ñàìîñòîÿòåëüíî).2. Íåïðåðûâíîñòü ôóíêöèè2.1. Îïðåäåëåíèå íåïðåðûâíîñòè ôóíêöèè. Ñâîéñòâàíåïðåðûâíûõ ôóíêöèéÎïðåäåëåíèå 2.1 Ôóíêöèÿ f (x) íàçûâàåòñÿ íåïðåðûâíîé âòî÷êå x0 , åñëè f (x) îïðåäåëåíà â îêðåñòíîñòè òî÷êè x0 èlim f (x) = f (x0 ).x→x0Îïðåäåëåíèå 2.2 Ôóíêöèÿ f (x) íàçûâàåòñÿ íåïðåðûâíîé íàìíîæåñòâå, åñëè îíà íåïðåðûâíà â êàæäîé òî÷êå ýòîãî ìíîæåñòâà.Òåîðåìà 2.1 Âñå ýëåìåíòàðíûå ôóíêöèè íåïðåðûâíû â îáëàñòèîïðåäåëåíèÿ.Òåîðåìà 2.2 Ïóñòü f (x) è g(x) íåïðåðûâíûå â òî÷êå x0 ôóíê-öèè. Òîãäà f (x) ± g(x), f (x) · g(x) òàêæå íåïðåðûâíûå â òî÷êåf (x)x0 ôóíêöèè. Åñëè g(x0 ) ̸= 0, òî- íåïðåðûâíàÿ â òî÷êå x0g(x)ôóíêöèÿ.71Òåîðåìà 2.3 Åñëè ôóíêöèÿ u = g(x) íåïðåðûâíà â òî÷êå x0 , àôóíêöèÿ y = f (u) íåïðåðûâíà â òî÷êå u0 = g(x0 ), òî ñëîæíàÿôóíêöèÿ y = f (g(x)) íåïðåðûâíà â òî÷êå x0 .2.2.
Îäíîñòîðîííèå ïðåäåëûÎïðåäåëåíèå 2.3 ×èñëî b íàçûâàåòñÿ ïðåäåëîì ôóíêöèèÊàÌôåäÃÒðÓàÌ ÂÌÈ-2ÐÝÀy = f (x) ïðè x ñòðåìÿùåìñÿ ê a ñïðàâà è îáîçíà÷àåòñÿlim f (x) = b (èëè lim f (x) = b),x→a+x→a+0åñëè äëÿ ëþáîãî ε > 0 íàéäåòñÿ òàêîå δ = δ(ε) > 0, ÷òî ïðèa < x < a + δ âûïîëíÿåòñÿ íåðàâåíñòâî |f (x) − b| < ε.Àíàëîãè÷íî îïðåäåëÿåòñÿ ïðåäåë f (x) ïðè x → a ñëåâàlim f (x) = b (ñ çàìåíîé íåðàâåíñòâà a < x < a+δ íà íåðàâåíñòâîx→a−0a − δ < x < a).Òåîðåìà 2.4 Ïðåäåë f (x) ïðè x → a ñóùåñòâóåò òîãäà è òîëü-êî òîãäà, êîãäà ñóùåñòâóþò îáà îäíîñòîðîííèõ ïðåäåëà è îíèðàâíû ìåæäó ñîáîé.
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