Учебно-методическое пособие (1017796), страница 2

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+ nn − n2 + 32 + 4 + 6 + . . . + 2n1 + 3 + 5 + . . . + (2n − 1)1 11+ + ... + n2 421111−+− . . . + (−1)n n5 25 1255√√√√39273n2 · 2 · 2 · ... · 21+1128291111+++ ... +1·3 3·5 5·7(2n − 1) · (2n + 1)ÊàÌôåäÃÒðÓàÌ ÂÌÈ-2ÐÝÀ3012 + 22 + 32 + . . . + n2n31111+++ ... +1·2 2·3 3·4n · (n + 1)Çàäà÷à 3. Èñïîëüçóÿ ëîãè÷åñêóþ ñèìâîëèêó, ñôîðìóëèðîâàòüîïðåäåëåíèå óêàçàííîãî ïðåäåëà. Äàòü ãåîìåòðè÷åñêóþ èíòåðïðåòàöèþ.1357911131517192123lim f (x) = ∞2lim f (x) = 44lim f (x) = −36lim f (x) = ∞8x→−∞x→∞x→0−x→∞lim f (x) = −∞ 10x→−2+0lim f (x) = ∞12lim f (x) = −314lim f (x) = +∞16lim f (x) = −∞18lim f (x) = −∞20lim f (x) = 022lim f (x) = ∞24x→1x→−3+0x→0x→0+x→−4x→∞x→+∞lim f (x) = +∞x→−1−0lim f (x) = −∞x→+∞lim f (x) = 0x→−∞lim f (x) = −1x→+∞lim f (x) = +∞x→∞lim f (x) = +∞x→−∞lim f (x) = −∞x→∞lim f (x) = +∞x→+∞lim f (x) = ∞x→−∞lim f (x) = −8x→0lim f (x) = −∞x→−3−0lim f (x) = −∞x→∞12252729lim f (x) = 026lim f (x) = −∞28lim f (x) = 030x→0+x→0x→+∞lim f (x) = −5x→−∞lim f (x) = +∞x→3+0lim f (x) = ∞x→2−0ÊàÌôåäÃÒðÓàÌ ÂÌÈ-2ÐÝÀÇàäà÷à 4.

Èñïîëüçóÿ ðàçëè÷íûå ïðèåìû ïðåîáðàçîâàíèÿ âû-ðàæåíèé, âû÷èñëèòü ïðåäåëû.1234567(3x + 7)3 (x − 2x2 )2limx→∞ (27x5 + 4x + 1)(2 − 4x)2(3x + 1)3 − (3x − 1)3limx→∞ (x + 1)2 + (x − 1)2√5 + 16x2lim √√√x→+∞ 416x2 + x + x√√6x x + 5 35x10 + 1lim√ √x→+∞ (x + 4 x) 3 x3 − 15x2 + 4x − 1lim √x→∞ 3 27x6 + 1 − 1√x + 5xlim √x→+∞ 3 x3 + 1 + sin x√√33x + 7 + 3 x2 + 2xlimx→∞17x + 4 cos 7x85x + 6xlimx→±∞ 5x − 6x9x2 + 5x + 6limx→−3 x2 + 4x + 3131011x2 − 7x + 6limx→1x4 − 1x4 − 1limx→1 2x4 − x2 − 1√√5x − 2 − 8limx→2x2 − 4√√1 + x2 − 1 − x2limx→03x2 − x4√x x−8lim √x→4x−2√4x−2lim √x→16x−4√39x − 3√lim √x→33 + x − 2x√9 + 2x − 5lim √3x→8x2 − 4√√325 + x − 3 29 − x√limx→2x − 2x√√3x2 − 2 3 x + 1limx→1(x − 1)2()2x − 1 1 + 2x2lim−x→∞ 5x + 72 + 5x2()112lim−x→2 2 − x8 − x3ÊàÌôåäÃÒðÓàÌ ÂÌÈ-2ÐÝÀ12x3 − 3x + 2limx→1 x4 − 4x + 3x3 − 3x − 2limx→−1 (x2 − x − 2)2131415161718192021222314(24limx→∞( 2))(2x − 1) 3x + x + 23x−2x + 14x2 − 12(25lim x +x→∞√31 − x3)√()33lim x − 4 − x26x→∞ÊàÌôåäÃÒðÓàÌ ÂÌÈ-2ÐÝÀ)(√2limx − 5x + 6 − x27x→±∞( (√))2lim x x + 1 − x282930x→+∞lim(√x→+∞limx→∞x4 + 8x2 + 3 −√)x4 + x2(√)3x3 − 2x2 + 3 − xÇàäà÷à 5.

Âû÷èñëèòü ïðåäåëû, èñïîëüçóÿ ýêâèâàëåíòíûå áåñ-êîíå÷íî ìàëûå1sin2 3xlimx→0 1 − cos x2cos 4x − 1x→0 x arctg x3(1 + tg x)10 − 1limx→0arcsin 5x4sin(x − 2)x→2 x3 − 85limarctg(x − 5)x→5 arcsin (x2 − 25)67(π)lim x −tg x2x→π/28limlimlim (x − 2) ctgx→2x−251 − cos(x − 3)x−3x→3(x − 3) tg2lim159limcos 2xx→π/4 4x − π10sin 3πxx→2 sin 8πx11tg 5xx→π sin 3x12sin 6πxx→1 sin πx13tg 2x − sin 2xx→0x314cos 5x − cos 3xx→0arctg x2151 − 2 cos x(limπ)x→π/3sin x −3162arcsin x − 1limx→0 3arctg x − 117e2x − e−2xlimx→0 ln(1 + 3 sin x)18e4x − ex)lim (x→0 ln 1 + 3 arcsin2 x1944x − 4xlimx→0 ln(cos x)20arcsin(x/(x + 1))x→0 ln (1 − 2x/(3 + x2 ))22ln(5 − 2x) − ln 5x→0arcsin 3x2423−x − 4lim √x→1 3 2 − x − 126ex−1 − 1lim √x→1x−128√31 + 6/x2 − 1limx→∞ cos(3/x) − 1limÊàÌôåäÃÒðÓàÌ ÂÌÈ-2ÐÝÀlimlimlim22122√41 + arctg2 x − 1limx→0sin2 3x23ln(3 − x) − ln 2limx→1sin(x − 1)257x−1 − 7limx→2 53−x − 527sin(1/x)limx→∞ tg(π/x)lim2limlim16( 1/x)(2 + 1) 2 − 1ln(cos(1/x))()29 lim30limx→+∞ arctg 3x tg2 (1/x)x→+∞ (3−x + 2) 31/x − 1−xÇàäà÷à 6.

Âû÷èñëèòü ïðåäåëû, èñïîëüçóÿ, ãäå ýòî âîçìîæíî,ÊàÌôåäÃÒðÓàÌ ÂÌÈ-2ÐÝÀâòîðîé çàìå÷àòåëüíûé ïðåäåë, ïðåäâàðèòåëüíî îáîñíîâàâ âîçìîæíîñòü åãî ïðèìåíåíèÿ(1alimx→+∞1c2a2b2c3a3b3c)x)x2x − 2limx→−∞ 3x + 4()x2x − 2limx→∞ 2x + 3)3/(x+1)( 2x −2limx→∞ x2 + 3( 2)x+1x −2limx→∞ x2 + 3)3x/(x+1)(x−2limx→∞ x2 + 3()x25lim 6 −x→0cos2 x()1/x25lim 6 −x→0cos2 x)1/x2(11lim 6 −x→02 cos2 x(1b2x − 23x + 417(4alimx→01 − cos 2x2x2)1/x2)1/x21 − cos 2xlimx→0x2()1/(x2 +1)1 − cos 2xlimx→02x2(4bÊàÌôåäÃÒðÓàÌ ÂÌÈ-2ÐÝÀ4c5a5b5clim (1 + sin x + x2 )1/xx→0lim (1 + x + x2 )1/(1+x)x→0lim (1 + cos x + x2 )−1/xx→0(6aπx )1/(x2 +1)lim cosx→036b(πx )1/x2lim cosx→03(6c7a7b7c2πx )2/(x2 +2)lim sinx→03()1/ sin x3 + 5xlimx→0 3 + 2x)−x2(3 + 5xlimx→0 3 + 2x()−1/x23 + 5x2limx→0 3 + 2x2218)1/x1 + 2xlimx→0 1 + 3x)(2x+3)/(2x+1)(1 + 2xlimx→0 1 + 3x()1/x21 + 2xlimx→0 1 + 3x()−1/x2sin 2xlimx→0x()−1/x2sin xlimx→0x()(2x+3)/(x+1)sin xlimx→0x(8a8bÊàÌôåäÃÒðÓàÌ ÂÌÈ-2ÐÝÀ8c9a9b9c(10a−x2lim 2 − 2x→∞(10b11a11b11c−x2lim 2 − 2)1/x2x→0(10c)−x2−x2lim 2 − 2)1/(x−1)2x→1()x2 arctg x1lim 2 − cosx→+∞x)x2 arctg x(1lim 2 − cosx→−∞xlim (2 − cos x)1/ arctgx→02x19(12a12bÊàÌôåäÃÒðÓàÌ ÂÌÈ-2ÐÝÀ12c)arctg2 (1/(x−2))4−xlimx→22()1/ arctg(x−2)4−xlimx→22()arctg2 (x−2)5−xlimx→22()−1/( √3 x−2)2x−7limx→8 x + 1()1/( √3 x−2)2x − 7limx→8x+113a13b13c lim (cos(x − 8))sin(x−8)/(x−8)x→814a( (π))1/x2lim tg−xx→06(14b14clim tg(πx→04lim (tg x)(2x+3)/(−2x+3)x→0(15a))(ex −1)/x2−xlimx→π/2x )1/ cos2 xtg315b( x )1/ cos xlim tg2x→π/215clim (tg x)π/(2x)x→π/220(16a16bÊàÌôåäÃÒðÓàÌ ÂÌÈ-2ÐÝÀ16c)1/xx + 2 sin xlimx→∞x)−1/x2(2x + sin xlimx→∞x()1/x2x + 2 sin xlimx→0x()3x2x − arctg xlimx→∞ 2x + arctg x()3x+22x − arctg xlimx→0 2x + arctg x()−3/(x−2)22 − arctg xlimx→2 2 + arctg x17a17b17c18a18b18c19a19b19c()1/ ln cos 2xlim 1 + tg2 xx→0()− ln2 xlim 1 + arctg2 xx→∞()ln cos 2xlim 1 + arctg2 xx→0lim (4 − 4x )(3x+sin x)/xx→02x 3x/ sin xlim (2 − 2 )x→02lim (2 − 2x )(3xx→−∞+sin x)/x21(20a20blim 3 − 3)5/(tg 5x sin 2x)x→π()3x2 / tg2 xlim 3 − 3sin xx→0(sin2 πxlim 3 − 3)−1/(x−1)2x→1ÊàÌôåäÃÒðÓàÌ ÂÌÈ-2ÐÝÀ20csin2 xÇàäà÷à 7.

Âû÷èñëèòü ïðåäåëû, èñïîëüçóÿ âòîðîé çàìå÷àòåëü-íûé ïðåäåë(1234567)xx−2limx→∞ x + 1)x+2(x−1limx→∞ x + 3)√x( 35x + 2limx→+∞5x3)2x( 2x − 2x − 3limx→∞ x2 + 2x + 1()1/ sin x3 + 5xlimx→0 3 + 2x()1/(2x)lim 1 + x + x2x→02lim (2 − cos x)1/xx→02289x→0(lim 6 −x→05cos2 x)1/x2()1/ ln cos 2xlim 1 + tg2 xx→0ÊàÌôåäÃÒðÓàÌ ÂÌÈ-2ÐÝÀ10lim (1 − sin 3x)1/ tg 5x1112lim (cos 2x)1/ tg143x22xx→0lim (cos x)1/ sinx→2π(132limx→01 + 2x1 + 3x)1/ sin xlim (1 − x sin 3x)1/ ln cos xx→0(1516)1/ sin 3xx+2limx→0 x2 + 2()√x2 +3√x + x2 + 1limx→+∞x(17limx→1(18limx→03x − 1x+1sin 2xtg 2x)1/(√x−1))1/(1−cos x)23(19lim tgx→0(π4(20))(ex −1)/x2−xx2lim 2 − e)1/(1−cos πx)x→02lim (2 − 2x )3x/ sin21xÊàÌôåäÃÒðÓàÌ ÂÌÈ-2ÐÝÀx→0(22limx→82x − 7x+1)1/( √3 x−2)x )1/ cos xlim tg2x→π/2()1/ ln(2−x)sin xlimx→1 sin 1(2324(2526(x−1)2lim 2ex→1()5/(tg 5x sin 2x)sin2 xlim 2 − 2x→π(2728limx→24−x2)2/ arctg(x−2)()x2 arctg x1lim 2 − cosx→+∞x(29)π2 / ln sin(πx/2)−1limx→12x − 1x)ln(3+2x)/ ln(3−2x)2430()sin(π/(2x)) ln(2−(x−1)/x)x−1lim 2 −x→∞xÇàäà÷à 8.

Èññëåäîâàòü íà íåïðåðûâíîñòü ôóíêöèþ f (x) è óêà-ÊàÌôåäÃÒðÓàÌ ÂÌÈ-2ÐÝÀçàòü õàðàêòåð òî÷åê ðàçðûâà.f (x)1|x + 3|·x+4x+323e1/(x+4)4579113x2 + x − 43x2 − x − 21arctg1−xπx sinxsin(1/x)x−π13 1 − (x + 2) sin15171921681012114x+2|x + 5|arctg(x + 5)()−11cos2x√6 − 32 + xx2 − 1611 − 3x/(1−x)16182022f (x)x+2· x2|x + 2|1(x + 3)(x − 5)5x2 − x − 62x2 − x − 3πsinx√13 + x − 4x2 − 9sin(1/x)πx − 11(x + 2) arctgxsin(1/x)ln |x|1e1/x − 11tgxxsin x25(23cos2√ )1/xx2421/x − 121/x + 111−x+3 x11−x x−31 1/xex2528/(16−x2 )xex(x+1) − 11 −1/x2exÊàÌôåäÃÒðÓàÌ ÂÌÈ-2ÐÝÀ27262ex2930x ln2 |x|Çàäà÷à 9.

Âû÷èñëèòü ïðîèçâîäíóþ y ′ (x).13579111315y(x)( ())x−1ln cosx( ( ))1arctg lnx( ())3ln ln 3 − 2x√2esin x√23 sin x +x2ln x5 cos2 x3√3(1 + ln x)442etg (3x)6y(x)(π)x · sin ln x −4xln2 x2ctg (1/x)8√cos2 x − 2 sin2 x2410121416)cos3 x (· 3 cos2 x − 515√1+ x√1− x1· arctg(3 tg x)3(√ )xarcsin226()arctg ex + e−x√x−119arctg22tg x21+ ln (cos x)2√2· (ln x − 5) 1 + ln x233√25ln3 (2x + 3)18√arcsin x − 120√tg 3x2ctg2 x ctg4 x22 ln(sin x) +−2422x + 1√ · arctg √2433√√326x+ xÊàÌôåäÃÒðÓàÌ ÂÌÈ-2ÐÝÀ1727291 + cos 2x1 − cos 2xtg5 x tg7 x+571ctg2 x21x2√· arcsin2328ln(sin x) +30Çàäà÷à 10.

Âû÷èñëèòü ïðîèçâîäíóþ y ′ (x).1357911y(x)1 + 2x2√x1 + x2√)1 + x2 ( 22x−13x3(√)√12√ ln3x + 3x − 232()2 5/2 5x − 11 + 2x·70()√1ln x + + x2 + x + 12(())√4 √44x3 − ln 1 + x3324681012√y(x))1 + x2 ( 2x −2( 3 √)2xxarctg++x−1242(1 + x)5/2 (5x − 2)35x−2√4 4x − x22(3x − 4)(2 + x)3/215√x+1−1ln √x+1+12715171921232527293x − 9 √3(x + 2)210√√1x2 − x + 2 + x − 2√ ln √√22x −x+2+x+ 2√√12 + 2x − 2 − x√ ln √√2 + 2x + 2 − x218x + 3arcsin √2 √411+xln√ tg x1 + 4xln 4sin3 x√20x + 32(x − 2) 4 x − 245√3)1 + x3 ( 2x−3x2()15 tg x + 4arctg33√3 x + 2lncos2 x1arctg2 √2xx−1arcsin √x()(x − 2)5ln(x + 1)34(3ex − 4)(ex + 1)3/4211x3ln24 x3 + 8141618ÊàÌôåäÃÒðÓàÌ ÂÌÈ-2ÐÝÀ13202224ln 5(1 + 3 tg x)262(10 + 3x)√9 5 + 3x28√2 x + 1(ln(x + 1) − 2)30Çàäà÷à 11. Âû÷èñëèòü ëîãàðèôìè÷åñêóþ ïðîèçâîäíóþ y ′ (x).1357y(x)√x x2 + 1√√33x + 1√4 x4 + 1(x − 1) 3 x2 + 1√x(2 − x)(sin x)1/x(tg x)cos x2√5x3y(x)x2(x2 + 1) (2x + 3)4xcos x6xtg x82(arcsin x)x2891113(cos x)x(sin 3x)x10121416√(x − 2)3(x − 1)5 (x − 3)11x(tg x)eÊàÌôåäÃÒðÓàÌ ÂÌÈ-2ÐÝÀ152x(1 + x )(arctg x)x√x−1√√3(x + 2)2 (x + 3)3√ √x3x−1x2 − 221819(ln sin 3x)x(cos x)sin x21x1/x221720(arctg x)ln x24()arccos x1 + x22526√4x2 + 3x + 1√2728√32 + 4 7x + 1x()10 ( 3)x6 x2 + 1x +12930(2x2 + 5)2233(1 + x3 )x√x x√√cos x · 2 cos x(sin x)ln x√ √√43x3 x2 x + 1√( 2) xx +1√3 3 1 + x√ 2x +12 1−xÇàäà÷à 12.

Âû÷èñëèòü ïðîèçâîäíóþ y ′ (x) ôóíêöèè, çàäàííîéïàðàìåòðè÷åñêè.1x(t), y(t)x=2 sin t1 + 3 cos t5 cos ty=1 + 3 cos t2x(t), y(t)() x = ln 1 + t2 y = t − arctg t29{3{x = e−ty = arctg(2t + 1)271 x = arccos √1 + t2t y = arcsin √1 + t29{6 x = t2 + 2t y = t2 − ln t2813101t2 − 1t2 + 1y=t+2{√x = √t3y= t y = 3 sin2 tt3t+1t2y=t+1x=12{14 x = 2 cos2 t15( )t x = ln tg+ cos t2 y = t sin t + cos tx=x=11x = 5(cos t + t sin t)y = 5(sin t − t cos t)ÊàÌôåäÃÒðÓàÌ ÂÌÈ-2ÐÝÀ5x = arcsin(t2 − 1)y = arccos 2t4( ) x = 4 tg2 t2y = 2 sin t + 3 cos t166t1 + t36t2y=1 + t3x = ety = arcsin t x = ln t()11t+y=2t3017x={212325276t1 (+ t2 )3 1 − t2y=1 + t218x=√arcsin ty = 1 − t220 x = 2t − t3 y = 2t2t3 + 2x=t2 + 1t3y= 2t +1)(1 x=2 t+t3 1 y =t+ +t t2 x = et cos t22 y = et sin t2824 y = t2 e−2t x = ln t26 y = sin2 tx=29t−1y=√1 + t2{x = 5 cos3 ty = 4 sin3 tcos3 tx=√cos 2tsin3 ty=√cos 2t x = t2 e−tÊàÌôåäÃÒðÓàÌ ÂÌÈ-2ÐÝÀ19√ x = 1 + t21t+1ty=t+130x=1t+1()2ty=t+1 x = t + ln cos t y = t − ln sin t31Çàäà÷à 13.

Âû÷èñëèòü ïðîèçâîäíóþ y ′ (x) ôóíêöèè, çàäàííîéíåÿâíî. F (x, y)ln x + e−y/x + 52x2/3 + y 2/3 − 10ÊàÌôåäÃÒðÓàÌ ÂÌÈ-2ÐÝÀ13√y − 4x − x2 + 10y − 4 + 34ex − ey + x − y − 65x−67√32x2 y 2 + 5x + y − 5 + 9√√3xy−−8=0√ −xyyx√yarctg − ln x2 + y 2 + 2x8ex − ey − xy9√y − 2x − x2 − 5xy − y10 2 cos2 (x + y) + xy − 9y √ 211 3 arctg − x + y 2 + 5xx12 ln 5y + + 7y13 x − arctg(x + y) + 132√14 y −32y − 1+ 12x15 ex sin y − e−y cos x16 x −√3x−y−6x+yÊàÌôåäÃÒðÓàÌ ÂÌÈ-2ÐÝÀ17 y 3 −y3 + x − 418 yex − 1 − ey + 919 y 2 − x − lny−4x√20 x + y − 3 3 x − y + 11√4y + 5321 y − x + 10y − 6 +x√x2 y322 x − 2y −+ −3yx√23 x − 1 + 2xy + y 2 − 8y 3 + 3x sin x−+3y sin y√√25x+y−y x−y−72426 ln(x + y) − √8x + y2()()27 sin y − x2 − ln y 2 − x28 ex + ey − 2xy − 2332y29 x + y 2 ln x − 430 x2 sin y + y 3 cos x − 2xÇàäà÷à 14. Íàïèñàòü óðàâíåíèÿ êàñàòåëüíîé è íîðìàëè ê êðè-1234567ÊàÌôåäÃÒðÓàÌ ÂÌÈ-2ÐÝÀâûì â çàäàííûõ òî÷êàõy = x4 − 6x2 + 8xy = arcsin(2x − 3)y=y=√35x − 3√3y=x(x − 2)2√31 − x2√y = x2 − 2xy=√sin xM1 (0; 0)()3M1;02M1 (−1; −2)M2 (1; 3)( π)M2 2;2()3M2;05M1 (1; 1)M2 (0; 0)M1 (3; −2)M2 (1; 0)√M1 (−1; 3))(π 1M1;√6 2M2 (2; 0)M2 (π; 0)y = xe1/(x−2)M1 (3; 3e)M2 (2 − 0; 0)y = x ln xM1 (1; 0)M2 (0 + 0; 0)10y = x2 ln xM1 (1; 0)M2 (0 + 0; 0)11y = xx8912y=sin xxM1 (1; 1)M2 (0 + 0; 1)(√ )π 2 2M1;M2 (0 + 0; 1)4 π3413yx+=116914x2 y 2−=1169√ )3 3M1 2;2()9M1 5;4153y 2 = x(x − 3)2M1 (3; 0)2(M2 (4; 0)M2 (4; 0)M2 (0; 0)()2 1M2 − ;e eÊàÌôåäÃÒðÓàÌ ÂÌÈ-2ÐÝÀ2162y ln y = x17x2/3 + y 2/3 = 118(x2 + y 2)2()= 2 x2 − y 2{1920{21222324x = 2t − t2y = 3t − t3{x = te3ty = te−2tx = t − sin ty = 1 − cos t{x = cos3 ty = sin3 t x= 1+tt331y=+2t2 2t(){x = ln 1 + t2y = t − arctg tM1 (0; 1)(√ )1 3 3M1;8 8(√)3 1M1;2 2M2 (0; 1)M2(√2; 0t1 = 0t2 = −1t1 = 0t2 = −t1 =π6t2 = 0t1 =π4t2 =13π232t1 = 1t2 = −t1 = 2t2 = 0)35{25√x = 1 − t2y = arcsin tt1 = −1r=φφ1 =27r = cos 2φφ1 =28r=√cos φφ1 =π2π4π3π3π3φ2 = 0φ2 = 0φ2 = 0ÊàÌôåäÃÒðÓàÌ ÂÌÈ-2ÐÝÀ26t2 = 029r = 1 − cos φφ1 =30r = 1 + cos φφ1 =φ2 = πφ2 = 0Çàäà÷à 15.

Óêàçàòü( ) òèï íåîïðåäåëåííîñòè. Ñâåñòè ê íåîïðå-äåëåííîñòè òèïàïðàâèëî Ëîïèòàëÿ.1357900(∞)èëè. Âû÷èñëèòü ïðåäåë, èñïîëüçóÿ∞ln xlimx→+∞ xln (1 + 3x )limx→+∞ ln (1 + 2x )()ln x2 + exlimx→+∞ ln (x4 + e2x )()ln 1 + x2(π)limx→+∞ln− arctg x2π/xlimx→0 ctg (πx/2)2()ln x2 + 3limx→∞ x4 + x2 + 14ln (1 + 3x )limx→−∞ ln (1 + 2x )6ln ctg xx→0+ ln x8ln xx→0+ 1 + 2 ln sin x10ln cos 2xx→π sin2 xlimlimlim36(112x − x2limx→2 2 − x1213tg πxx→1 4x − 41415( x)2lim e − x16limlimx→011− xx e −1limx→+∞)(√ )ln x − xln tg xlim πx→π/4−x4(√)ln 1 + 4x − 218 limx→2x−2ÊàÌôåäÃÒðÓàÌ ÂÌÈ-2ÐÝÀx→+∞1719lim ln x · ln(x − 1)x→1+0lim (2 − x) · ln ln(2 − x) 20x→2−0lim x ln2 x2123252729ln tg xx→π/4 1 − ctg xlimlim x3 e−x22x→0+πxlim (1 − x) tgx→1224x· ln(π − x)x→π−02()11lim−x→1 x − 1ln x()xπ−lim2 cos xx→π/2 ctg xlim cos2628x→+∞x2xlimx→+∞ 3x()11lim−x→0 sin xx()11lim−x→0 tg xsin x1 1/xex→0− x230limÇàäà÷à 16.

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