Теория пределов и числовые ряды (Конспект), страница 3

Êðàòêî åå ÷èòàþò òàê: ñóììà íåñêîëüêèõáåñêîíå÷íî ìàëûõ ôóíêöèé åñòü ôóíêöèÿ áåñêîíå÷íî ìàëàÿ.Ôóíêöèÿ y = x1 + √1x + x12 ÿâëÿåòñÿ áåñêîíå÷íî ìàëîéôóíêöèåé ïðè x → ∞, òàê êàê êàæäîå ñëàãàåìîå √1x , x1 è x12 åñòüáåñêîíå÷íî ìàëàÿ ôóíêöèÿ ïðè x → ∞.Ïðèìåð 7.4.Ôóíêöèÿ y = x+x3 +x5 åñòü áåñêîíå÷íî ìàëàÿ ôóíêöèÿ ïðè x → 0, òàê êàê ôóíêöèè y = x, y = x3 è y = x5 áåñêîíå÷íîìàëûå ïðè x → 0.Ïðèìåð 7.5.Òåîðåìà 7.2. Ïðîèçâåäåíèå áåñêîíå÷íî ìàëîé ôóíêöèè ïðè x → ∞íà ôóíêöèþ, îãðàíè÷åííóþ ïðè x → ∞, ÿâëÿåòñÿ ôóíêöèåé áåñêîíå÷íî ìàëîé.|φ(x)| 6 C ïðè x > N îãðàíè÷åííàÿ ôóíêöèÿ;|α(x)| 6 Cε ïðè x > N - áåñêîíå÷íî ìàëàÿ ôóíêöèÿ.Äîêàçàòü, ÷òî γ(x) = φ(x) · α(x) áåñêîíå÷íî ìàëàÿ ôóíêöèÿx → ∞, ò.å.

|γ(x)| < ε ïðè x > N .Äàíî:Äîêàçàòåëüñòâî:|γ(x)| = |φ(x) · α(x)| = |φ(x)| · |α(x)| < C ·÷òî è òðåáîâàëîñü äîêàçàòü.ε= ε,CïðèËåêöèÿ 7. Òåîðèÿ ïðåäåëîâ. Îñíîâíûå òåîðåìû117xÔóíêöèÿ y = cosÿâëÿåòñÿ áåñêîíå÷íî ìàëîé ïðèx4x → ∞, òàê êàê îíà ÿâëÿåòñÿ ïðîèçâåäåíèåì îãðàíè÷åííîé ôóíêöèècos x íà áåñêîíå÷íî ìàëóþ (ïðè x → ∞) ôóíêöèþ y = x14 .Ïðèìåð 7.6.Ôóíêöèÿ y = x(1 + sin x) ÿâëÿåòñÿ áåñêîíå÷íî ìàëîé ïðè x → 0, òàê êàê îíà ÿâëÿåòñÿ ïðîèçâåäåíèåì îãðàíè÷åííîéôóíêöèè 1 + sin x íà ôóíêöèþ x, áåñêîíå÷íî ìàëóþ ïðè x → 0.Ïðèìåð 7.7.Ñëåäñòâèå.1. Òàê êàê âñÿêàÿ áåñêîíå÷íî ìàëàÿ ôóíêöèÿ îãðàíè÷åíà, òî èç òîëüêî ÷òî äîêàçàííîé òåîðåìû âûòåêàåò, ÷òî ïðîèçâåäåíèåäâóõ áåñêîíå÷íî ìàëûõ ôóíêöèé åñòü ôóíêöèÿ áåñêîíå÷íî ìàëàÿ.Ñëåäñòâèå.2.

Ïðîèçâåäåíèå áåñêîíå÷íî ìàëîé ôóíêöèè íà ÷èñëîåñòü ôóíêöèÿ áåñêîíå÷íî ìàëàÿ.×àñòíîå îò äåëåíèÿ ôóíêöèè α(x), áåñêîíå÷íî ìàëîé ïðè x → a, íà ôóíêöèþ φ(x), ïðåäåë êîòîðîé (ïðè x → a) îòëè÷åí îò íóëÿ, ÿâëÿåòñÿ ôóíêöèåé áåñêîíå÷íî ìàëîé.Òåîðåìà 7.3.Äîêàçàòåëüñòâî: Ôóíêöèÿα(x)ìîæåò áûòü ïðåäñòàâëåíà â âèäåφ(x)ïðîèçâåäåíèÿ áåñêîíå÷íî ìàëîé ôóíêöèè α(x) íà îãðàíè÷åííóþ ôóíêα(x)1. Íî òîãäà èç òåîðåìû 7.2 âûòåêàåò, ÷òî ÷àñòíîå= α(x) ·öèþφ(x)φ(x)1ÿâëÿåòñÿ áåñêîíå÷íî ìàëîé ôóíêöèåé.φ(x)7.2.

Áåñêîíå÷íî áîëüøèå ôóíêöèèÎïðåäåëåíèå 7.2. Ôóíêöèÿ y = N (x) íàçûâàåòñÿ áåñêîíå÷íîáîëüøîé ïðè x → a, åñëè äëÿ ëþáîãî ïîëîæèòåëüíîãî ÷èñëà L ìîæíîïîäîáðàòü òàêîå ÷èñëî δ > 0, ÷òî äëÿ âñåõ çíà÷åíèé x ∈ (a − δ; a + δ)âûïîëíÿåòñÿ íåðàâåíñòâî |N (x)| > L.Òàê, íàïðèìåð, ôóíêöèÿx → ∞.y = x2ÿâëÿåòñÿ áåñêîíå÷íî áîëüøîé ïðèÊàêîå áû ïîëîæèòåëüíîå ÷èñëîìîæåò áûòü ñäåëàíà áîëüøå, ÷åìLLìû íè âçÿëè, ýòà ôóíêöèÿ(äëÿ âñåõ çíà÷åíèéÑèìâîëè÷åñêè áåñêîíå÷íî áîëüøàÿ, ïðèx → ax>N =√L).ïîëîæèòåëüíàÿôóíêöèÿ çàïèñûâàåòñÿ â âèäå:lim N (x) = ∞.x→aÅñëè áåñêîíå÷íî áîëüøàÿ ôóíêöèÿ îòðèöàòåëüíà, òî ãîâîðÿò, ÷òî îíàñòðåìèòñÿ ê−∞è ïèøóò:lim N (x) = −∞.x→a118Ëåêöèÿ 7.

Òåîðèÿ ïðåäåëîâ. Îñíîâíûå òåîðåìûÁåñêîíå÷íî áîëüøèå ôóíêöèè ïðèíÿòî îáîçíà÷àòü ëàòèíñêèìè áîëü-øèìè áóêâàìèN (x), M (x).7.3. Ñâÿçü áåñêîíå÷íî áîëüøèõ è áåñêîíå÷íî ìàëûõôóíêöèéÌåæäó áåñêîíå÷íî áîëüøèìè è áåñêîíå÷íî ìàëûìè ôóíêöèÿìèñóùåñòâóåò òåñíàÿ ñâÿçü, êîòîðàÿ óñòàíàâëèâàåòñÿ â ñëåäóþùèõ òåîðåìàõ.Åñëè ôóíêöèÿ N (x) ÿâëÿåòñÿ áåñêîíå÷íî áîëüøîéïðè x → a, òî ôóíêöèÿ N 1(x) áåñêîíå÷íî ìàëàÿ ïðè x → a.Òåîðåìà 7.4.Äîêàçàòåëüñòâî: Ïóñòüx → ∞ Âîçüì¼ì ïðîèçâîëüíîå ε > 0.

Ïîêàx âûïîëíÿåòñÿ íåðàâåíñòâî | N 1(x) | < ε,æåì, ÷òî äëÿ äîñòàòî÷íî áîëüøèõ1 áåñêîíå÷íî ìàëàÿ ôóíêöèÿ. Òàê êàê ïîN (x)óñëîâèþ N (x) áåñêîíå÷íî áîëüøàÿ ôóíêöèÿ, òî ñóùåñòâóåò òàêîå11÷èñëî C , ÷òî |N (x)| >ïðè x > C . Íî òîãäà || < ε äëÿ òåõ æå x.εN (x)Òåì ñàìûì òåîðåìà äîêàçàíà.à ýòî è îçíà÷àåò, ÷òîÔóíêöèÿ y = x2 áåñêîíå÷íî áîëüøàÿ ïðè x → ∞.Ñëåäîâàòåëüíî, ôóíêöèÿ x12 ÿâëÿåòñÿ áåñêîíå÷íî ìàëîé ïðè x → ∞.Ïðèìåð 7.8.òîÅñëè ôóíêöèÿ α(x) åñòü áåñêîíå÷íî ìàëàÿ ïðè x → a, áåñêîíå÷íî áîëüøàÿ ôóíêöèÿ ïðè x → a.Òåîðåìà 7.5.1α(x)Òåîðåìà ïðèâîäèòñÿ áåç äîêàçàòåëüñòâà.7.4. Îñíîâíûå òåîðåìû î ïðåäåëàõÍèæå ïðèâîäÿòñÿ îñíîâíûå òåîðåìû î ïðåäåëàõ, êîòîðûå ïîçâîëÿþò îáëåã÷èòü îïðåäåëåíèå ïðåäåëîâ.

Ïðè ýòîì ôîðìóëèðîâêè è äîêàçàòåëüñòâà òåîðåì äëÿ ñëó÷àåâx → ∞, x → −∞, x → x0 , x → x0 − 0, x → x0 + 0ñîâåðøåííî àíàëî-ãè÷íû. Ïîýòîìó çäåñü îíè ïðåäëàãàþòñÿ äëÿ îáùåãî ñëó÷àÿx → a.Òåîðåìà 7.6. Åñëè ôóíêöèÿ y = f (x) èìååò ïðåäåë (ïðè x → a),ðàâíûé b, òî å¼ ìîæíî ïðåäñòàâèòü â âèäå ñóììû ÷èñëà b è áåñêîíå÷íî ìàëîé ôóíêöèè α(x) ïðè x → a:f (x) = b + α(x).(7.2)Ëåêöèÿ 7.

Òåîðèÿ ïðåäåëîâ. Îñíîâíûå òåîðåìûÄàíî:lim α(x) = 0x→aèìååò ïðåäåë.119 áåñêîíå÷íî ìàëàÿ,lim f (x) = bx→af (x) = b + α(x).lim f (x) = b ⇒ |f (x) − b| < ε,Äîêàçàòü, ÷òî ïðè ýòîìÄîêàçàòåëüñòâî:x→alim (f (x) − b) = 0,ýòî çíà÷èò, ÷òîx→aìàëàÿ ôóíêöèÿ, ïðèx→aò.å.ïðèf (x) − b = α(x)ñëåäîâàòåëüíî, ôóíêöèÿ|x − a| < δ ,à áåñêîíå÷íîf (x) = b + α(x).Òåîðåìà 7.7. (îáðàòíàÿ, áåç äîêàçàòåëüñòâà). Åñëè ôóíêöèþy = f (x) ìîæíî ïðåäñòàâèòü êàê ñóììó ÷èñëà b è íåêîòîðîé áåñêîíå÷íî ìàëîé ôóíêöèè (ïðè x → a), òî ÷èñëî b ÿâëÿåòñÿ ïðåäåëîìôóíêöèè f (x) (ïðè x → a).Òåîðåìà 7.8. Ïðåäåë ñóììû (ðàçíîñòè) äâóõ ôóíêöèé ðàâåí ñóììå (ðàçíîñòè) èõ ïðåäåëîâ.Åñëèlim φ(x) = blim ψ(x) = c, òî ôóíêöèè f (x)x→aòîæå èìåþò ïðåäåëû ïðè x → a.x→af (x) = φ(x) − ψ(x)è= φ(x) + ψ(x)lim [φ(x) ± ψ(x)] = lim φ(x) ± lim ψ(x).x→aÄàíî:x→ax→a(7.3)lim φ(x) = b, lim ψ(x) = c.x→aÄîêàçàòü, ÷òîx→alim [φ(x) + ψ(x)] = lim φ(x) + lim ψ(x).x→aÄîêàçàòåëüñòâî:lim φ(x) = b,x→ax→a}{íà îñíîâàíèè òåîðåìû7.6 ⇒ áåñêîíå÷íî ìàëûå ïðèx → a.x→alim ψ(x) = c;x→aãäåèα(x)èβ(x)φ(x) = b + α(x),ψ(x) = c + β(x).Òîãäàf (x) = φ(x) + ψ(x) = [b + α(x)] + [c + β(x)] = (b + c) + [α(x) + β(x)],lim f (x) = lim [φ(x) + ψ(x)] = lim {(b + c) + [α(x) + β(x)]} = b + c.x→ax→ax→aÏîñëåäíåå ðàâåíñòâî âûòåêàåò èç òåîðåìû 7.7.Ñëåäîâàòåëüíî:lim f (x) = lim [φ(x) + ψ(x)] = lim φ(x) + lim ψ(x).x→ax→ax→ax→aÀíàëîãè÷íî äîêàçûâàåòñÿ, ÷òîlim [φ(x) − ψ(x)] = lim φ(x) − lim ψ(x).x→ax→ax→a120Ëåêöèÿ 7.

Òåîðèÿ ïðåäåëîâ. Îñíîâíûå òåîðåìûÒåîðåìà 7.9. Ïðåäåë ïðîèçâåäåíèÿ äâóõ ôóíêöèé ðàâåí ïðîèçâåäåíèþ èõ ïðåäåëîâ.lim φ(x) = b è lim ψ(x) = c, òî ôóíêöèÿx→ax→aòàêæå èìååò ïðåäåë ïðè x → a, ïðè÷¼ìÅñëèf (x) = φ(x) · ψ(x)lim [φ(x) · ψ(x)] = lim φ(x) · lim ψ(x).x→aÄàíî:x→a(7.4)x→alim φ(x) = b, lim ψ(x) = c.x→aÄîêàçàòü, ÷òîx→alim [φ(x) · ψ(x)] = lim φ(x) · lim ψ(x).x→aÄîêàçàòåëüñòâîlim φ(x) = b,x→ax→a}x→alim ψ(x) = c;{íà îñíîâàíèè òåîðåìû7.6 ⇒x→aφ(x) = b + α(x),ψ(x) = c + β(x),α(x) è β(x) áåñêîíå÷íî ìàëûå ïðè x → a.f (x) = φ(x) · ψ(x) = [b + α(x)] · [c + β(x)] = (b · c) + [c · α(x) + b · β(x) ++ α(x) · β(x)].ãäålim f (x) = lim [φ(x)·ψ(x)] = lim {(b·c)+[c·α(x)+b·β(x)+α(x)·β(x)]} =x→ax→ax→a= b · c + lim [c · α(x) + b · β(x) + α(x) · β(x)] = b · c = lim φ(x) · lim ψ(x).x→aÇäåñüx→ax→alim [c·α(x)+b·β(x)+α(x)·β(x)] = 0, ò.ê. âñå ñëàãàåìûå áåñêîíå÷íîx→aìàëûå ôóíêöèè.Ñëåäñòâèå 1.

Ïîñòîÿííûé ìíîæèòåëü ìîæíî âûíîñèòü çà çíàê ïðåäåëà, ò.å.lim [k · φ(x)] = k · lim φ(x).x→ax→a(7.5)Ñëåäñòâèå 2. Èç òåîðåìû 7.9 âûòåêàåò, ÷òî ïðåäåë ñòåïåíè ðàâåíñòåïåíè ïðåäåëà:lim [f (x)]n = [lim f (x)]n .x→ax→a(7.6)Òåîðåìû 7.8 è 7.9 ðàñïðîñòðàíÿþòñÿ íà ëþáîå êîíå÷íîå ÷èñëî ôóíêöèé.Òåîðåìà 7.10. (áåç äîêàçàòåëüñòâà). Ïðåäåë äðîáè ðàâåí îòíîøåíèþ ïðåäåëà ÷èñëèòåëÿ è çíàìåíàòåëÿ, åñëè ïîñëåäíèé íå ðàâåííóëþ.Åñëèlim φ(x) = b, lim ψ(x) = cx→ax→aèc ̸= 0,òîlim [φ(x)/ψ(x)] = lim φ(x)/ lim ψ(x).x→ax→ax→a(7.7)Ëåêöèÿ 7.

Òåîðèÿ ïðåäåëîâ. Îñíîâíûå òåîðåìûÏðèìåð 7.9.Íàéòè lim (x2 + 2x − 1).Ð å ø å í è å:•121x→2Âîñïîëüçóåìñÿ òåîðåìàìè î ïðåäåëàõ ôóíêöèè.Ïî òåîðåìå î ïðåäåëå ñóììûlim (x2 + 2x − 1) = lim x2 + lim 2x − lim 1.x→2•x→2x→2x→2Òàê êàê ïðåäåë ñòåïåíè, ðàâåí ñòåïåíè ïðåäåëà, òîlim x2 = [lim x]2 = 22 = 4.x→2•x→2Ïîñòîÿííûé ñîìíîæèòåëü ìîæíî âûíîñèòü çà çíàê ïðåäåëàlim 2x = 2 lim x = 2 · 2 = 4.x→2•x→2Ïðåäåë ïîñòîÿííîé ðàâåí ñàìîé ïîñòîÿííîélim 1 = 1.x→2Îêîí÷àòåëüíî ïîëó÷èì:lim (x2 + 2x − 1) = lim x2 + lim 2x − lim 1 = 4 + 4 − 1 = 7.x→2x→2Ïðèìåð 7.10.x→2x→2x4 +3x2 +4.2x→1 x −2x+3Íàéòè limÐ å ø å í è å: Íà îñíîâàíèè íàâûêîâ, ïðèîáðåò¼ííûõ ïðè ðåøåíèèïðåäûäóùåãî ïðèìåðà, íàõîäèì:4242ïðåäåë ÷èñëèòåëÿ lim (x +3x +4) = lim x +lim 3x +lim 4 = 1 + 3 + 4 = 8,x→1x→1x→1x→122ïðåäåë çíàìåíàòåëÿ lim (x −2x+3) = lim x −lim 2x+lim 3 = 1 − 2 + 3 = 2.x→1x→1x→1x→1Ïðèìåíÿÿ òåîðåìó î ïðåäåëå äðîáè, ïîëó÷èìlim (x4 + 3x2 + 4)x4 + 3x2 + 48x→1lim 2=== 4.x→1 x − 2x + 3lim (x2 − 2x + 3)2x→1(î ïðîìåæóòî÷íîé ôóíêöèè).

Ïóñòü äàíû òðèôóíêöèè, óäîâëåòâîðÿþùèå íåðàâåíñòâàì φ(x) 6 f (x) 6 g(x) äëÿäîñòàòî÷íî áëèçêèõ ê a çíà÷åíèé x. Åñëè ôóíêöèè φ(x) è g(x) èìåþòîäèí è òîò æå ïðåäåë ïðè x → a, òî è ôóíêöèÿ f (x), çàêëþ÷åííàÿìåæäó íèìè, èìååò ïðåäåë, ðàâíûé ïðåäåëó ôóíêöèé φ(x) è g(x).Òåîðåìà 7.11. êà÷åñòâå äîêàçàòåëüñòâà ïðèâåäåì ïðîñòóþ ãåîìåòðè÷åñêóþ èíòåðïðåòàöèþ óñëîâèé òåîðåìû ïðèx→∞(ðèñ.