11. , - , . . () , - . , . . .2. . , , , , . . . : . , , . , . , -, Bolzano - Weierstrass . , . , : - , .3. . ; , ,, .4. . n0 . , . ( ) , , .5. - : , , i . . . , ,, , . . , y= xn, log x =x11tdt . , , ., , , . . ,: (i) (log x=x11tdt) , (ii) ( ) ( ) , (iii) ( arctan x=x01t2+1 dt) (iv) ( ) . , .6. Riemann Darboux. . Darboux Riemann .7. . , . ; ! ,, ; .8. - . , . . .9. -ii 2007 - 08 2008- 09 . ; . : , , . , ., , . 2009.iiiiv 2 . ( ) . ( ). . limx (x=) , x , limx , x . . . . 2010. 3 , . , -. . 2011.vviCalculus , , ,T. Apostol.Dierential and Integral Calculus,R. Courant.Introduction to Calculus and Analysis,R. Courant - F. John.A Course of Pure Mathematics,G. Hardy.A Course of Higher Mathematics,V. Smirnov.Advanced Calculus (Schaums Outline Series),M. Spiegel.The Calculus A Genetic Approach,O. Toeplitz.viiviii1 . 11.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.1 ,,,. . . . . . . . . . . . . . . . . . . . . . . 11.1.2 , . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.3 R. . . . . . . . . . . . . . . . . . . . . 21.1.4 . . . . . . . . . . . . . . . . . . . . . . . 31.1.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.2.1 . . . . . . . . . . . . . . . . . . . . . . . . . 61.2.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.2.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.2.4 . . . . . . . . . . . . . . . . . . . . . . . . . 111.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.4 . . . . . . . . . . . . . 161.4.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161.4.2 . . . . . . . . . . . . . . . . . . . . . 182 . 232.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332.4 . . . . . . . . . . . . . . . . . . . . . . . . . 382.4.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382.4.2 . . . . . . . . . . . . . . . . . . . . . . . . . . 382.4.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 462.4.4 . . . . . . . . . . . . . . . . . . . . . . . . 482.5 . e, . . . . . . . . . . . . . . . . . . . . . 533 . 613.1 . . . . . . . . . . . . . . . . . . . . . . . . . 613.2 , , . . . . . . . . . . . . . . . . . . . . . . . 633.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 663.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 673.4.1 ,,. . . . . . . . . . . . . . . . . . . . . . . . 673.4.2 . . . . . . . . . . . . . . . . . . . . 763.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 793.6 . . . . . . . . . . . . . . . . . . . . . . . . . . 833.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85ix3.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 873.9 . . . . . . . . . . . . . . . . . . . . . . . . . 893.10 . . . . . . . . . . . . 903.10.1 . . . . . . . . . . . . . . . . . . . . . . . . . . 903.10.2 . . . . . . . . . . . . . . . . . . . 933.11 . . . . . . . . . . . . . . . . 963.11.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 963.11.2 . . . . . . . . . . . . . . . . . . . . . 974 . 1014.1 ,. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1014.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1154.3 . . . . . . . . . . . . . . . . . . . . . . . . 1194.3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1194.3.2 . . . . . . . . . . . . . . . . . . . . . . . . . . 1204.3.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1274.3.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1284.3.5 . . . . . . . . . . . . . . . . . . . . . . . . 1304.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . 1344.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1364.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1394.7 . . . . . . . . . . . . . . . . . . . . . . . . . 1414.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1444.9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1485 . 1515.1 ,. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1515.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1555.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1605.4 . . . . . . . . . . . . . . . . . . . . . . . . . . 1635.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1655.6 . . . . . . . . . . . . . . . . . . . . . . . . 1715.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1766 . 1816.1 . . . . . . . . . . . . . . . . . . . . . . . 1816.1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1816.1.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1826.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1836.3 ,. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1866.4 . . . . . . . . . . . . . . . . . . . . . . . . . 1886.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1926.5.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1926.5.2 . . . . . . . . . . . . . . . . . . . . . . . . 1936.5.3 . . . . . . . . . . . . . . . . . . . . . . 1936.5.4 . . . . . . . . . . . . . . . . . . . . . . . 1956.5.5 . . . . . . . . . . . . . . . . . . . . . 1966.6 ,. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2016.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . 203x6.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2086.9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2156.9.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2156.9.2 . ,. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2186.10 . . . . . . . . . . . . . . . . . . . . . . . . . . 2246.10.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2256.10.2 . . . . . . . . . . . . . . . . . . . . . . . . . 2256.10.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2286.10.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2296.10.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2316.11 . . . . . . . . . . . . . . . . . . . . . . . . . 2376.11.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2376.11.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2406.12 , . . . . . . . . . . . . . . . . . . . . . . . . . 2446.12.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2446.12.2 . . . . . . . . . . . 2467 Riemann. 2517.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2517.2 Riemann. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2577.3 Riemann. . . . . . . . . . . . . . . . . . . . . . . . . . 2627.3.1 . . . . . . . . . . . . . . . . . . . . . 2627.3.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2647.3.3 . . . . . . . . . . . . . . . . . . 2657.3.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . 2677.3.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2697.4 Riemann. . . . . . . . . . . . . . . . . . . . . . . . . 2747.4.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2747.4.2 . . . . . . . . . . . . . . . . . . . . . . 2757.4.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2807.4.4 . . . . . . . . . . . . . . . . . . . . . . . . . 2847.4.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2868 Riemann. 2918.1 Riemann. . . . . . . . . . . . . . . . . 2918.1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2918.1.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2938.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2978.3 Riemann. . . . . . . . . . . . . . . . . . . . . . . . 3048.3.1 . . . . . . . . . . . . . . . 3048.3.2 . . . . . . . . . . . 3058.3.3 . . . . . . . . . . . . . . . . . . . . . . 3068.3.4 . . . . . . . . . . . . . . . . 3128.3.5 . . . . . . . . . . . . . . 3178.4 Riemann. . . . . . . . . . . . . . . . . . . . . . . . . . 3258.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3298.5.1 . . . . . . . . 3318.5.2 . . . . . . . . . . . . . . . . . . . . . . 332xi8.5.3 . . . . . . . . . . . . . . . . . . . . . . 3358.6 ,. . . . . . . . . . . . . . 3408.6.1 . . . . . . . . . . . . . . 3408.6.2 . . . . . . . . . . . 3429 . 3479.1 Taylor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3479.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 3509.3 Riemann. . . . . . . . . . . . . . . . . . . . . . . . . 3529.3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . 3539.3.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 3549.3.3 . . . . . . . . . . . . . . . . . . . . . . . . . . 3549.3.4 Simpson. . . . . . . . . . . . . . . . . . . . . . . . . . . . 35510 . 35710.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35710.2 - . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36010.3p- . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36610.3.1 p- . . . . . . . . . . . . . . . . . . . . . . . . . . 36610.3.2 p- [0, 1). . . . . . . . 36910.3.3 p- - . . . . . . . . . . . . . . . . . . 37310.3.4 p- . . . . . . . . . . . . . . 37410.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37710.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38110.6 Taylor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38810.7 ,. . . . . . . . . . . . . 39410.7.1 . . . . . . . . . . . . . . 39410.7.2 , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39610.7.3 . . . . . . . . . . . . . . . . . . . . . . . . 398xii 1 . . . . , , , .. .1.1 . . ,,. x +y, x y, xy xy(y ,= 0) x, y. (, .) .1.1.1 ,,,. 1, 2, 3,. . . , 0, 1, 1, 2, 2, 3, 3,. . . , mn , m, n n ,= 0. : mn ,2m2n ,3m3n , . mn m n , . ,43 43 43 43 . , m m1 , . . , . , , , . . ; , .R. , N, Z Q , , . : 0, N 0, 1, 2,. . . N 1, 2,. . . .1.1.2 , . . .11.1. (1) x y y z, x z. , .(2) x y,x + z y + z x z y z.(3) x y z w, x + z y + w. , .(4) x y z> 0,xz yzxz yz .(5) x y z< 0,xz yzxz yz .(6) 0x + 33x + 1 , (x 2)2 4,[x27x[ > x27x,(x 1)(x + 4)(x 7)(x + 5)> 0,(x 1)(x 3)(x 2)2 0.52. x x .(, 3], (2, +), (3, 7), (, 2) (1, 4) (7, +),[2, 4] [6, +), [1, 4) (4, 8], (, 2] [1, 4) [7, +).. .1. x [x] = [x] ;2. k , [x + k] = [x] + k.(: [y] =m m m y< m + 1.)3. [x + y] = [x] + [y] [x + y] = [x] + [y] + 1 . [x + y + z].4. ,0 < x 1, n 1n+1< x 1n n x.5. b n>b. n b; b n b;a>0. n1n 0 a ,= 0 , n ,(i)an> 0 a > 0 (ii)an< 0 a < 0.1.5 .6 1.5. n 2,xnyn= (x y)(xn1+ xn2y + + xyn2+ yn1)., n 3,xn+ yn= (x + y)(xn1xn2y + xyn2+ yn1). n 12(n 1)n n! . n! = 12(n 1)n..1! = 1, 2! = 12 = 2, 3! = 123 = 6, 4! = 1234 = 24.,0! = 1 n n! = (n 1)! n. _nm_ m, n0 m n _nm_ =n!m!(n m)! .. _n0_ =_nn_ = 1, _n1_ =_nn1_ = n, _n2_ =_nn2_ =n(n1)2.1 m n,,,_nm_ =n(n 1)(n m + 1)m!. 1.6. Newton. x, y n (x + y)n=_n0_xn+_n1_xn1y + +_nn 1_xyn1+_nn_yn.. Newton . n = 1 (x + y)1=_10_x1+_11_y1, _10_ =_11_ = 1. n x + y. (x + y)n+1=_n0_xn+1+_n1_xny + +_nn_xyn+_n0_xny + +_nn1_xyn+_nn_yn+1. _n0_=1=_n+10_ _nn_=1=_n+1n+1_. , m-(1 m n) _nm_xnm+1ym+_nm1_xnm+1ym=_n+1m_xnm+1ym( 4 ) ,, (x + y)n+1=_n+10_xn+1+_n+11_xny + +_n+1n_xyn+_n+1n+1_yn+1. n + 1 n.7. (x + y)1= x + y ,(x + y)2= x2+ 2xy + y2,(x + y)3= x3+ 3x2y + 3xy2+ y3,(x + y)4= x4+ 4x3y + 6x2y2+ 4xy3+ y4,(x + y)5= x5+ 5x4y + 10x3y2+ 10x2y3+ 5xy4+ y5 Newton. 1.7 . 1.7. (1) .axbx= (ab)x, axay= ax+y, (ax)y= (ay)x= axy.(2) 0 < a < b,(i)ax< bx,x > 0, (ii)a0= b0= 1 (iii)ax> bx,x < 0.(3) x < y,(i)ax< ay,a > 1, (ii)1x= 1y= 1 (iii)ax> ay,0 < a < 1.. 1.7 ; ! , .(1) : x > 0,axbx= aa. .xbb. .x= (ab)(ab). .x= (ab)x.x < 0,a, b ,= 0 axbx=1aa. .x1bb. .x=1(ab)(ab). .x= (ab)x.x = 0,a, b ,= 0 axbx= 11 = 1 = (ab)x. : x, y> 0,axay= aa. .xaa. .y= aa. .x+y= ax+y. x < 0 < y x+y> 0, 0 < x < y, ay= aa. .y= aa. .xaa. .y(x)= aa. .xaa. .x+y= aa. .xax+y,, axay=1aa. .xay= ax+y. x0 n. , a < 0 na n 2. n = 2, 3, 4,. . . , na , , , . . . a. n=22a a , , a. n = 3 3a a..(1) x4=16 , 416=2 416= 2.,x4= 16 .(2) x5= 32 ,532 = 2. x5= 32 , 532 = 2. . 1.9. n, k. nk k n-.. . k n- , m k = mn. nk =nmn= m ,,., r=nk r=ml , m, l > 1. l >1, p l. l, m > 1, p m. k = rn=mnln, lnk=mn. p l, lnk, , mn. , , . ,p mn=m m, m . l = 1,r = m ,,k = rn= mn n- ..(1) 2 , , mm2= 2. : 12< 2 22> 2. ,35 m m3= 5.(2) 2+3 . , r, (2+3)2= r2, 6 =r252,, 6 . m m2= 6.91.2.3 . ar r . 1.1. a > 0, m, k n, l mn=kl . (na)m= (la)k.. ((na)m)nl= (na)mnl= ((na)n)ml= aml, ((la)k)nl= (la)knl= ((la)l)kn= akn. ml =kn, aml=akn, ((na)m)nl=((la)k)nl. , (na)m>0 (la)k>0, (na)m= (la)k., a>0 r. m nr=mn . , 1.1, (na)m ., ar= (na)m., r > 0 0r= 0. , 0r, 0r= (n0)mr =mn> 0 , m, n . , n a1n=na a 0. , ar, ar> 0 a > 0 r., a 0, (ii) a = 0 r > 0 (iii) a < 0 r.ar (i) a = 0 r 0 (ii) a < 0 r ..(1)234= (42)3, 268= (82)6= (42)3, 234= (42)3=1(42)3 , 262= (22)6=23= 8 262= (22)6=1(22)6=18 .(2)034= 0 035= 0. 034 , 035 00 .(3) (2)0= 1 2102= 25= 32. (2)53 (2)52 . , - 1.7, . 1.7 . 1.7 , . x =mn y=kl . (ax)n= am : (ax)n= ((na)m)n= ((na)n)m= am.(1) : a, b >0. (axbx)n=(ax)n(bx)n=ambm=(ab)m=((ab)x)n, axbx, (ab)x> 0, axbx= (ab)x.a = 0, x > 0 axbx= 0bx= 0 = 0x= (ab)x. b = 0 .: x + y=pq , p=ml + kn q=nl. a>0, (axay)nl=(ax)nl(ay)nl=((ax)n)l((ay)l)n=(am)l(ak)n=amlakn=aml+kn=((qa)q)ml+kn=(qa)q(ml+kn)=(qa)pnl=((qa)p)nl=(ax+y)nl,axay, ax+y> 0, axay= ax+y.10a = 0, x, y> 0 axay= 00 = 0 = ax+y.: xy=pq , p=mk q=nl. a>0, ((ax)y)nl=_((ax)y)l_n=((ax)k)n=((ax)n)k=(am)k=amk=((qa)q)mk=(qa)qmk=(qa)pnl=((qa)p)nl=(axy)nl, (ax)y, axy>0, (ax)y= axy. (ay)x= axy .a = 0,x, y> 0 (ax)y= 0y= 0 = axy. (ay)x= axy .(2)(i) x > 0, m > 0. (ax)n= am< bm= (bx)n,ax, bx> 0, ax< bx.(iii) (ii) .(3) (i) x0, ml ay.(iii) (ii) .1.2.4 ., axa 0 x . a > 1. , s, r, t s 0. ; x3+ x2y + xy2+ y3>0 x5+ x4y + x3y2+x2y3+ xy4+ y5> 0; ;3. n (i)1 + 2 + + n =12 n(n + 1),(ii)12+ 22+ + n2=16 n(n + 1)(2n + 1),(iii)13+ 23+ + n3=14 n2(n + 1)2.124. Newton Pascal;11 11 2 11 3 3 11 4 6 4 11 5 10 10 5 1 1 m n, _n+1m_=_nm_+_nm1_. Pascal;5. _nm_ n m ; _nm_ m 0 n; Pascal;6. n ._n0_+_n1_+ +_nn 1_+_nn_ = 2n,_n0__n1_+ + (1)n1_nn 1_+ (1)n_nn_ = 0.. .1. n a 0, nan= a. n , nan= [a[.2. a + b a +b a, b 0. a + b = a +b a = 0 b = 0.3. :nanb =nab ,nma =mna =nma n, m. ;4. n 0 a < b, na 0, aa aa=a2na n .138. a.(: 1 0 ,= 1. : y ax= y( x) ; x ax>0, , ax=y, y>0. 1.4, , , , y. 1.4. a > 0 a ,= 1. y> 0 x ax= y. 1.4 ax= y. . , x1, x2 ax= y( y) 1.7 ,x1 ,= x2 ,ax1,= ax2. a = 1, ax= y, . , 1x= 1 x, y 1 : . a=0 . 0x=y y=0 14 : . a < 0 ax x . ax= y y a logay. , :x = logay ax= y. 1.10. a > 0 a ,= 1.(1) loga(yz) = logay + logaz y, z> 0.(2) logayz= logay logaz y, z> 0.(3) loga(yz) = z logay y> 0 z.(4) loga 1 = 0 logaa = 1.(5) 0 < y< z. (i) logay< logaz,a > 1, (ii) logay> logaz,0 < a < 1.. (1) x= loga y w= loga z, ax=y aw=z. ax+w=axaw=yz, loga(yz) = x + w = loga y + loga z.(2) logayz+ loga z= loga(yzz) = loga y logayz= loga y loga z.(3) x = loga y,ax= y. azx= (ax)z= yz,,loga(yz) = zx = z loga y.(4) loga 1 = 0 a0= 1 loga a = 1 a1= a.(5) 0 < y< z. x = loga y w= loga z, y= ax z= aw. ax< aw,a > 1, x < w,0 < a < 1, x > w. 1.11. a, b > 0 a, b ,= 1. logby=1logab logay y> 0.. a, b>0 a, b ,=1. x= logb y w= loga b, bx=y aw=b. awx= (aw)x= bx= y. loga y= wx = loga b logb y..1. log2 4,log122,log124.2. log2 3 log3 4.3. log2 3log3 5log5 7log7 10log10 8.4. a > 0, a ,= 1. aloga y= y y> 0.5. log2 3 ;6. a > 0, a ,= 1. log1ay= logay y> 0.7. a > 0, a ,= 1. logaz (yz) = logay y> 0 z ,= 0.151.4 . -.1.4.1 . 1 , ( ) ( ). x [x[, , x>0, ( ) x < 0. x, : x , . - 1. , 2 B, , 32 2. x [0, 2] x 2 . x [k2, (k + 1)2], k : 2. , x, x ( ) 2. 1.5: . . x ,, 1. . cos x = +, , , .2. . sin x = 16+, , , .3. o . tan x = +, , , .4. o . cot x = +, , , . tan x , ,x =2+ k(k Z). , cot x ,,x = k(k Z). (cos x, sin x) . cos x, sin x, tan x cot x, ,, , x. x. , ..(1) cos 0 = 1,sin 0 = 0,tan 0 = 0. cot 0.(2) cos2= 0,sin2= 1,cot2= 0. tan2 .(3) cos = 1,sin = 0,tan = 0. cot .(4) cos32= 0,sin32= 1,cot32= 0. tan32. cos x >0, x (2+k2,2+k2)(k Z), cos x < 0, x (2+k2,32+k2)(k Z)., sin x > 0, x (k2, +k2)(k Z), sin x < 0, x ( + k2, 2 + k2)(k Z).,, 1 cos x 1, 1 sin x 1 x. 1.12 ; . 1.12. (1)(sin x)2+ (cos x)2= 1.(2) tan x =sin xcos x ,cot x =cos xsin x .(3) cos(x) = cos x,sin(x) = sin x,tan(x) = tan x,cot(x) = cot x.(4) cos(2 x) = sin x,sin(2 x) = cos x,tan(2 x) = cot x,cot(2 x) = tan x.(5) cos(x + ) = cos x,sin(x + ) = sin x,tan(x + ) = tan x,cot(x + ) = cot x.(6) cos(x + y) = cos x cos y sin x sin y, sin(x + y) = sin xcos y + cos x sin y.(7) cos x cos y= 2 sinxy2sinx+y2,sin x sin y= 2 sinxy2cosx+y2.(8) k. (i) cos x > cos x

, k2 x < x

+ k2, (ii) cos x < cos x

, + k2 x < x

2 + k2.(9) k . (i) sin x < sin x

, 2 +k2 x < x

2 +k2, (ii) sin x > sin x

,2+ k2 x < x

32+ k2.17. 1.12 .(1) .(2), = , , tan x=sin xcos x . , , = ,,cot x =cos xsin x .(3) x, x .(4) x,2 x .(5) x, x + .(6) , x, y x+y. ( ) ( ) . , ,(cos(x + y) 1)2+ (sin(x + y) 0)2=(cos x cos(y))2+ (sin x sin(y))2. , (1) (3), (6). , (3) (4):sin(x + y) = cos_2 (x + y)_ = cos__2 x_+ (y)_ = cos_2 x_cos(y) sin_2 x_sin(y)= sin xcos y + cos xsin y.(7) cos x = cos_x+y2+xy2_ = cosx+y2cosxy2sinx+y2sinxy2cos y= cos_x+y2xy2_ = cosx+y2cosxy2+ sinx+y2sinxy2.,cos x cos y= 2 sinx+y2sinxy2. (7).(8) k2 x < x

+k2, , x, x

, cos x > cos x

. +k2 x < x

2 +k2, , , cos x < cos x

.(9) 2+ k2 x < x

2+ k2, , x, x

, sin x < sin x

. 2+k2 x < x

32+k2, , , sin x > sin x

1.4.2 .. . .1. y[1, 1]. y . arccos y [0, ] , .2. y [1, 1]. y . arcsin y [2,2] .183. y. o y. . arctan y (2,2) .4. y. o y. . arccot y (0, ) ..(1) arccos 1 = 0,arccos 0 =2 ,arccos(1) = .(2) arcsin 1 =2 ,arcsin 0 = 0,arcsin(1) = 2 .(3) arctan 0 = 0 arccot 0 =2 . arccos y, arcsin y, arctan y arccot y , , -, -, - -y. y. y[1, 1] arccos y [0, ] cos x = y. x = arccos y cos x = y 0 x . cos x=y[, 0], arccos y. cos x = y R arccos y + k2(k Z) arccos y + k2(k Z)., , y[1, 1] arcsin y [2,2] sin x = y. x = arcsin y sin x = y 2 x 2 . sin x = y [2,32], arcsin y. sin x = y R arcsin y + k2(k Z) arcsin y + k2(k Z). y arctan y (2,2) tan x = y. x = arctan y tan x = y 2< x arccos y

arcsin y< arcsin y

.(2) y< y

. arctan y< arctan y

arccot y> arccot y

.19. (1) , y, y

, . , M , , arccos y> arccos y

., , y, y

, . , , , arcsin y< arcsin y

.(2) , y, y

, . , ,, arctan y< arctan y

., y, y

, . , ,, arccot y> arccot y

. . ( ) . ,,- . , . . : cos x sin x 8 10. , , . . 1. 2 . 8 10... .1. 6 ,43 .2. .cos x =12 , sin x = 12 , cos x = 12, sin x =32,tan x = 0, cot x = 1, tan x = 3 , cot x =3 .3. [a cos x + b sin x[ a2+ b2.4. a, b a2+ b2= 1 q [0, 2) cos q= a sin q= b.5. a, b 0. , p > 0 q a cos x + b sin x = p cos(x q) x.(: a cos x + b sin x =a2+ b2_aa2+b2 cos x +ba2+b2 sin x_.)206. (i) cos y= cos x y= x + k2y= x + k2(k Z),(ii) sin y= sin x y= x + k2y= x + k2(k Z),(iii) tan y= tan x y= x + k(k Z),(iv) cot y= cot x y= x + k(k Z).7. 1 + (tan x)2=1(cos x)2 , 1 + (cot x)2=1(sin x)2 .8. tan(x + y) =tan x + tan y1 tan xtan y , cot(x + y) = cot x cot y 1cot x + cot y.9. cos(2x) = (cos x)2(sin x)2= 2(cos x)21 = 1 2(sin x)2, sin(2x) = 2 sin xcos x,tan(2x) =2 tan x1 (tan x)2 , cot(2x) =(cot x)212 cot x.10. cos x =1 (tanx2)21 + (tanx2)2, sin x =2 tanx21 + (tanx2)2 , tan x =2 tanx21 (tanx2)2 , cot x =1 (tanx2)22 tanx2.11. 2 sin x sin y= cos(x y) cos(x + y), 2 cos x cos y= cos(x y) + cos(x + y),2 sin xcos y= sin(x y) + sin(x + y).12. , cos x + cos(2x) + cos(3x) + + cos(nx) =sin(nx2) cos((n+1)x2)sinx2,sin x + sin(2x) + sin(3x) + + sin(nx) =sin(nx2) sin((n+1)x2)sinx2.(: sinx2 .). .1. - 0, 12 , 22, 32 1.2. arccos y + arcsin y=2(1 y 1), arctan y +arccot y=2.213. y y= cos(arccos y) y= sin(arcsin y); y y= tan(arctan y) y= cot( arccot y);4. arccos(cos x) = x x [0, ]., arccos(cos x), x [k, (k +1)], k ; arcsin(sin x), arctan(tan x) arccot(cot x);22 2 . (). , . : n0 , . . . . . . e .2.1 . ( ) : , , . ; 1, 2, 3 . :x1, x2, . . . , xn,. . ., y1, y2, . . . , yn,. . ., z1, z2, . . . , zn,. . .. (xn), (yn), (zn). n (, -) , : , . xn+1 xn xn1 xn ..(1) (1n), 1,12 ,13 , . . . ,1n ,. . . .(2) (n), 1, 2, 3, 4, . . . , n,. . . .(3) (1), 1, 1, 1, . . . , 1,. . . .(4) _(1)n1_, 1, 1, 1, 1, . . . , 1, 1,. . . .(5) _110n_, 110 ,1102 ,1103 , . . . ,110n ,. . . .(6) n- n, 1, 2, 2, 3, 2, 4, 2, 4, 3, 4, 2,. . . . . : ,23 . , : n-. (, ) . -, , ( , ). , . - 1 . , , ; 1, 1, 1,. . . . , , - (, , ) ( ).(xn) xn+1xn n. (xn) xn+1> xn n. (xn) xn+1 xnn. , (xn) xn+1< xn n. . : , .(xn) , c xn= c n. . .(xn) , u xn u n. , (xn) , l l xn n. (xn) , l, u l xn u n, , [l, u]..(1) (c) , , . [c, c].(2) (1n) [0, 1].(3) _(1)n1n_ [12, 1].(4) (n1n) [0, 1].(5) _(1)n1_ [1, 1].(6) _(1+(1)n1)n2_, 1, 0, 3, 0, 5, 0, 7, 0,. . . , . 0 . , , u . (1+(1)n1)n2 u n,, n = 2k 1,2k 1 u k. , k u+12 k, 1.1.(7) 1, 0, 3, 0, 5, 0, 7, 0,. . . , , 24 ; l , l, l, .(8) _(1)n1n_, 1, 2, 3, 4, 5, 6,. . . , . u l (1)n1n u nl (1)n1n n, , n,. (xn) , xn [l, u]. (xn) 0. [M, M] [l, u], M= maxu, l. , (xn) , M [xn[ Mn. u (xn), > u (xn). , , . , l (xn), 0, () , 1=p+22=p2. , + = a 1+2= b . xn= 1n1+ 2n1 n 1.(ii) =p2+ 4q=0, , =p2 . , = a + = b . xn= n1+ (n 1)n1 n 1.(iii) =p2+4q 1000000132; 7576>1000000132( 7575 1000000132) ,, n 7576, 1n < 0, 000132. , 0, 0000000000132. , n 75757575758, 1n < 0, 0000000000132. , , : ( 0, 000132 0, 0000000000132) (7576 75757575758). , , . . , , . , , , n0 , , nn0 , 1n 0, , n0 1n0,1n0+1 ,1n0+2 ,. . . 1 1, >83 (2 2 3 = 2 + 1 = [2] + 1.29(1

0) n0 n0=[1

] + 1. , ( ) n0 , n0 .. (xn) x xx (xn) [xn x[ > 0 n n0, , >0n0 [xnx[ 0 n0 [xnx[ 0 n0 n n0 [xnx[ < ,, > 0 n0 n n0 [xnx[ < . 2.3: x < xn< x + n n0 . (xn) , (xn) . xn x ; n, n(, ), xn x xn , , x n ..(1) ,1n 0 n1n 1.(2) _(1)n1_. 1, 1, 1, 1,. . ., 1, 1, 1, 1,. . .. n , (1)n1 1 1 1 ., _(1)n1_ . (, , ) 1 (,, ) 1. _(1)n1_ ., , _(1)n1_ ., , _(1)n1_ x. >0 n0 [(1)n1 x[ < n n0 . n0 n n0 n n0 . , n n0 [ 1 x[ 0).> 0 n0 [1na 0[ < n n0 . [1na 0[ (1

)1a . (1

)1a 0, n0=[(1

)1a] + 1, n n0 n > (1

)1a,, [1na 0[ < . :1n2 0,1n 0,13n 0. xn x , . >0 , [xnx[ a . : 1 2 , , 1 = 2. n>an0=[a] + 1, a 0, n0=1, a < 0, n n0 n > a , , [xnx[ < ..(1) 1n2+n 0. > 0 n0 [1n2+n 0[ 12 +121 +4

n < 12121 +4

(: n ) n > 12 +121 +4

. 12 + 121 +4

0. , , n0=_12 + 121 +4

+1, n n0 n > 12+121 +4

,, [1n2+n 0[ < . . [1n2+n 0[ 0. [xn0[ 00; 1 .(3) sin nn 0. >0. [sin nn 0[ 0. [an0[ 1, n n0 n > log|a| ,, [an0[ < . a > 1. ..1. (xn), xn=3+(1)n2n, - . xn> xn+1 n xn< xn+1 n 3.2. . n0 .limn+34 , limn+n83, limn+1nn , limn+3n4n , limn+(1)n8n32n.3. :n 23n + 4 13 ,3nn + 3 2,nn + 1 0. ; , n- 32 n ., , n( , n = 1000, 10000 ). n0 .4. (, >0 n0 , ), .1n 0,1n5 0,110n 0,_13_n 0,1n + 1 0,3n 14n + 5 34 ,1n2+ 1 0,1n + 1 0,n2n + 13n2+ 213 ,2n + 32 3n 23 ,cos nnn 0,cos(2n) +nn 1,12n+ 3n 0.5. xn x, (xn) x, : >0 n0 n n0 [xnx[ .;2.3 ..(1) (n2) 1, 4, 9, 16, 25, 36, 49, 64,. . .. n- n2 n . , n2 n .(2) (n) 1, 2, 3, 4, 5, 6, 7, 8,. . .. n- n n . (xn) :xn n ., , , . :xn n .33. (n2) : n2 n . ; , , 35000. n2 35000n ; , : n n2 35000; : n2>35000 n>35000 . n >35000 ; 188 >35000( 187 35000),, n 188, n2 > 35000. , , 35000000000. , n187083, n2 >35000000000. , ( 35000 35000000000) ( 188 187083), . , M, , n0 ,, n n0 ,n2 > M. n2> M n >M . ,, 1.1: M>0 n0 n0, n0 + 1, n0 + 2,. . . >M . ( M) n0 M. n0 ( M) n >M ;: M 0, n0 n0= [M] + 1. . (xn) + + + (xn) xn M> 0 n n0,, M> 0 n0 xn> Mn n0 . (xn) + xn + limxn= + limn+xn= +. . (xn) + M>0 n0 xn>M n n0,, M>0 n0 n n0 xn>M,, M> 0 n0 n n0 xn> M. 2.4: xn> M n n0 ., (xn) (xn)xn M 0 n0 xn< Mn n0 . (xn) xn limxn= limn+xn= .34 xn + xn ; xn + xn, , ,, n ..(1) , n2 + n +.(2) n< M n>M. , M> 0 n0 n < M n n0 M>0 n0 n2< Mn n0 . n2 n .(3) (xn), xn=_3(1)n1_n2=_n, n 2n, n , -, 1, 4, 3, 8, 5, 12, 7, 16,. . . . xn +. M>0. xn>M(xn n) n>M. , n0=[M] + 1, n n0 n > M,, xn> M. (xn) +. 1 .(4) _(1)n1_ .2.6, + ; 1 1. , , , , _(1)n1_ +. M>0 n0 (1)n1>M n n0 . , n0 (1)n1> 1 n n0 . ,, ! _(1)n1_ ., _(1)n1_ ; , +, . limn+(1)n1 .(5)(n) (n2). a>0(na), 1, 2a, 3a, 4a,. . . . :na + (a > 0). M> 0 n0 n n0 na> M. na> M n > M1a . , n0= [M1a] + 1, n n0 n > M1a,, na> M. : n2 +,5n +.(6) a > 1 (logan), loga 1 = 0, loga 2, loga 3, loga 4,. . . . :logan + (a > 1). M>0 logan>M n > aM. n0= [aM] + 1, n n0 n > aM,, logan > M.(7) . a >1 (an). +. 35 2, 22, 23, 24,. . . 10, 102, 103, 104. . . . +.,,M> 0 an> M n > logaM. , logaM 0, M 1, logaM< 0, M< 1. , n0= [logaM] +1, M 1, n0= 1, M< 1, n n0 n > logaM,, an> M. , 2.6, a 1, (an) ; 1 1. , + ,, . :an___ +, a > 1 1, a = 1 0, 1 < a < 1 , a 1(8) _(1)n1n_, 1, 2, 3, 4, 5, 6, 7,. . . , ; (a 1), 1 1.(9) n2+nn+3 +. M>0. n2+nn+3>M n2+ n>Mn + 3M n2+ (1 M)n 3M>0 n>M12+12(M 1)2+ 12MnM12+12(M 1)2+ 12M . M12+12(M 1)2+ 12M 0. , n0=_M12+12(M 1)2+ 12M+ 1, n n0 n >M12+12(M 1)2+ 12M,,n2+nn+3> M. . n2+nn+3>M(n2+nn+3n2n+3n=n4) n4>M n>4M. n0= [4M] + 1, n n0 n > 4M,,n2+nn+3> M.(10) n7+ 2n2n +. M> 0. n7+2n2n > M (n7+2n2n n7) n7>M n>7M . n0=_7M + 1, n n0 n>7M, , n7+ 2n2 n>M. . n7+ 2n2 n>M (n7+ 2n2 n 2n2) 2n2>M n>M2. n0=_M2+ 1, n n0 n >M2,, n7+ 2n2n > M. , lim, limn+ .: , . + (-) (-) . -, 36 + . , , , . +, . + . , . , (!!), x x . , . , ..1. (xn) xn=(3(1)n1)n2. xn> xn+1 n xn< xn+1 n.2. , , . M n0 .limn+n2n , limn+log3n, limn+22n3n, limn+(3)2n, limn+(3)3n.3. x ,= 1 , , limn+(1x)n(1+x)n .4. M n0 , + . ( ) (;) n.(n218n 4),_7n 130n2_,_n30n + 1_,_1 n1 +n_,_n2+ 1n + 100_.5. (, M>0 n0 M, ), .n4 +, 3n , 3n +, log2_1n_ , n22n +,n252n + 1 +, n + (1)n1 +, 2n5+ n3 ,n + 2n +, n(2 + sin n) +.6. xn +, (xn) +, : M>0 n0 n n0 xn M.;xn , (xn) , : M>0 n0 n n0 xn M.;372.4 .2.4.1 . 1,12,13,14,15,16,. . ., 2, 5,14,15,16,. . .. (1n) 0. 0. 2.1. , . . (xn) xn n ,, . ,, ( ,), .. (xn) (yn) . , N, M xN=yM , xN+1=yM+1 , xN+2=yM+2. , , xn . yn . >0, n0

[xn [0, n0

xn>M2 n n0

n0

yn>M2 n n0

. n0=maxn0

, n0

, n0 n0

n0 n0

. xn>M2 yn>M2 n n0 . xn + yn>M2+M2= M n n0,, xn + yn + = (+) + (+).xn + yn y. M>0, n0

xn>M y + 1 n n0

n0

[yn y[ < 1 n n0

. [yn y[ < 1 yn>y 1 n n0

. n0=maxn0

, n0

, n0 n0

n0 n0

. xn>M y+ 1 yn>y 1 n n0 . xn + yn> (M y + 1) + (y 1) = M n n0,, xn + yn + = (+) + y. .39.(1) 1n 0 (1)nn 0. 1n+(1)nn 0 + 0 = 0.(2) n + 1n 0. n +1n (+) + 0 = +.(3) n n . n n () + () = . +, :.(1)n + 3 (+) + 3 = +, n (n + 3) + (n) = 3 3., 3 .(2)2n +, n 2n + (n) = n +.(3)n +, 2n n + (2n) = n .(4) n + (1)n1 n 1 n n 1 +. 2.4.3 ( 2.3), n+(1)n1 +. , n _n + (1)n1_+ (n) = (1)n1 . (+) x = +, x () = +, (+) () = +,() x = , x (+) = , () (+) = , . (+) (+), () () .+() - () () x x (, x). (xn), (yn), (xnyn), , , , xnyn= xn + (yn). . (xn), (yn) limn+xnlimn+yn , (xnyn) limn+(xnyn) = limn+xn limn+yn . .()x = , x() = (x > 0),()x = , x() = (x < 0),()() = +, ()() = . ()0, 0() . (+)(+) + : , , 40 . (+)x = + (x > 0) : x (, x) , , . .(+)0 (, 0); . : (i)2100 , 1250 2100

1250= 250 , (ii) 250 ,12100 250

12100=1250 , (iii) 2100 , 12100 2100

12100= 1 . (xn), (yn) (xnyn) n- n- . . (xn), (yn) limn+xn limn+yn , (xnyn) limn+xnyn= limn+xnlimn+yn .. xn x yn y. >0, n0

[xn x[M n n0

. n0= maxn0

, n0

, n0 n0

n0 n0

. xn>M yn>M n n0 . xnyn>MM= M n n0,, xnyn + = (+)(+).xn + yn y>0. M>0, n0

xn>2My n n0

n0

[yn y[2My [yn y[2Myy2=Mn n0,, xnyn + = (+)y. . ..(1) 1n 0 (1)nn 0. (1)nn2=1n(1)nn 00 = 0.(2) n1n 1 1n 0. n1n2=n1n1n 10 = 0.(3) n + 1n . nn2= n(1n) (+)() = . n n2 .(4) . (xn) c. limn+xn climn+xn ,limn+cxn= c limn+xn(c ,= 0 limn+xn= ). (xn) (c) c.41(5) a > 0.c > 0,cna c(+) = +.c < 0,cna c(+) = .(6) a > 0,cna c0 = 0.(7) n. a0 +a1x + +aNxN (aN ,= 0 N 1). a0 + a1n + + aNnN aN(+) =_+, aN> 0, aN< 0 aNnN ,a0 + a1n + + aNnN= aNnN_a0aN1nN+a1aN1nN1+ +aN1aN1n+ 1_, 1, 0. limn+(a0 +a1n + +aNnN) = limn+aNnN 1 = aN(+). n . limn+_a0 + a1n + + aNnN_ = limn+aNnN. : 3n25n + 2 + 12n5+ 4n4n3 .(8) . a (1 +a +a2+ +an1+an), 1 +a, 1 +a +a2, 1 +a +a2+a3,. . . . :1 + a + a2+ + an___ +, a 111a , 1 < a < 1 , a 1 .a > 1,1 + a + a2+ + an=an+11a1(+)1a1= +.a = 1,1 + a + a2+ + an= n + 1 +. 1 < a < 1,1 + a + a2+ + an=1an+11a101a=11a ., a 1. xn= 1 +a +a2+ +an an+11 = (a 1)(1 +a +a2+ +an) an+1= (a 1)xn +1. limn+(1 +a +a2+ +an) =limn+xn , limn+an+1= (a1) limn+xn+1. , limn+an+1 , limn+(1 + a + a2+ + an) . , ()k k:(+)k= (+)(+). .k= +, ()k= ()(). .k= .()k= +, k , , k . 2.2. limn+xnk,42limn+xnk= ( limn+xn)k.,limn+xnk= limn+(xn xn. .k) = limn+xn limn+xn. .k= (limn+xn)k..(1) n1n 1. (n1n)3 13= 1.(2) n52n2+ n 7 +. (n52n2+ n 7)8 (+)8= +.(3) 2n3+ n2+ 2n 7 . (2n3+ n2+ 2n 7)4 ()4= +.(4) n3+ 2n 1 . (n3+ 2n 1)5 ()5= . +, 0, ..(1)n +,7n 0 n 7n= 7 7., 7.(2)n2 +,1n 0 n2

1n= n +.(3)n +,1n2 0 n 1n2=1n 0.(4)n +,(1)n1n 0 n (1)n1n= (1)n1 . 1+= 0,1= 0. 10 . 1 0 : ,, ( 0) (, 0).10 . 0 (, 0) .,, , . , , . (xn) (1xn) , , xn ,= 0 n., . xn ,=0 n. (xn) 1limn+xn (, limn+xn ,= 0), _1xn_ limn+1xn=1limn+xn.43. xnx >0. >0, n0 [xnx[ < minx2

2,x2 n n0 . [xnx[ 0),x= (x < 0),x= 0. x0 ,0, , . x0= x10 10 . 0= () 10 10 . = () 1= ()0 .+() () () x x (, x). (xn) (yn), (xnyn), , xnyn= xn

1yn. . yn ,=0 n. (xn), (yn) limn+xnlimn+yn , _xnyn_ limn+xnyn= limn+xnlimn+yn.44.(1) n. a0+a1x++aNxNb0+b1x++bMxM , aN ,= 0, bM ,= 0. a0 + a1n + + aNnNb0 + b1n + + bMnM ___aNbM (+), N> MaNbM, N= M0, N< M,a0 + a1n + + aNnNb0 + b1n + + bMnM=aNbMnNMa0aN1nN+a1aN1nN1+ +aN1aN1n+ 1b0bM1nM+b1bM1nM1+ +bM1bM1n+ 1, , 1. limn+a0+a1n++aNnNb0+b1n++bMnM=aNbM limn+nNM

11=aNbM limn+nNM . , , n . limn+a0 + a1n + + aNnNb0 + b1n + + bMnM= limn+aNnNbMnM . ,:n32n2+n+12n23n1 +,n2+nn+2 ,n4n37n4+n+1 1,n2+n+4n3+n2+5n+6 0.(2)_2n3+n2+n+12n+3_7 ()7= .(3)_n3+n+73n3+n2+1_3 (13)3= 127 ., 00++ ..(1)2n 0,1n 0 2n1n= 2 2. 2 .(2)1n 0,1n2 0 1n1n2= n +.(3)1n2 0,1n 0 1n21n=1n 0.(4)5n +, n + 5nn= 5 5. 5 .(5)n2 +, n + n2n= n +.(6)n +, n2 + nn2=1n 0. [ +[ = +, [ [ = +. [ [ =+ : - , , .,, ([xn[) (xn).45 . (xn) , ([xn[) limn+[xn[ =limn+xn.. xn x. > 0, n0 [xn x[ Mxn< M,, n n0 . [xn[ > M n n0,, [xn[ + = [ [..(1) (5n) 4, 3, 2, 1, 0, 1, 2, 3,. . . 5n . ([5 n[) 4, 3, 2, 1, 0, 1, 2, 3,. . . [5 n[ [ [ = +.(2) . (1)n1 = 1 1. , limn+(1)n1.2.4.3 . 2.3. xn yn n.(1) xn +,yn +.(2) yn ,xn .. (1) M>0. xn +, n0 xn>Mn n0,yn xn , yn> M n n0 . , yn +.(2) (1)..(1) n+(1)n1 n1 n1 + n + (1)n1 +.(2) n2+2n+1n+2 n n + n2+2n+1n+2 +.(3) [n] > n 1 n 1 + [n] +. 2.4. xn yn n. xn x yn y,x y.. ( ) x > y. =xy2> 0, xn x yn y n0

[xn x[ l ,, (xn) l. , (3) , limn+xn u l,l < u, .47.(1) a 1, (an) , 1 1.(2) _(1)n1n_ , 1 1.(3) _n 3[n3]_ , n 3[n3] = 0 n = 3k(k Z) n 3[n3] = 1 n = 3k + 1(k Z).2.4.4 . (1n), _(1)n1n_ (n1n) , , -. 2.7. 2.7. , .. xn x. = 1, n0 [xnx[ < 1 n n0 . [xnx[ < 1 [xn[ = [(xnx)+x[ [xnx[+[x[ < 1+[x[. [xn[ < 1 +[x[ n n0 . M= max[x1[, . . . , [xn01[, 1 +[x[, [xn[ Mn.. 2.7. H _(1)n1_ .2.8. (1) +, .(2) , .. (1)xn +. M=1, n0 xn>1 n n0 . l= minx1, . . . , xn01, 1 xn l n. , u, n0 xn> u n n0,,u (xn). (xn) .(2) , , (1)..(1)(1)2.8. _(1+(1)n1)n2_, 1, 0, 3, 0, 5, 0, 7,. . . , ., + 0.(2) 1, 0, 3, 0, 5, 0, 7,. . . , ... . .1. (xn) . i xn+1= xn + 2, xn+3= xn3, xn+1= xn23, xn+2= xn2+ 3,xn+1= xn2+ 3, xn+2= xn+1 + xn3.(: , .)482. , ,,-, .limn+_2n3+ 3n +1n_, limn+_n +1n+ 2(1)n1n_, limn+n2n + 3n,limn+_1 +1n_3, limn+_n +(1)n1n_9, limn+(n + 1)27(n + 3)79(2n + 1)106,limn+1 +1n22 +(1)n1n, limn+n +1n2 +(1)n1n, limn+1 +(1)n1nn + 3 log10n ,limn+n + (1)nn2 + (1)n, limn+11n+1n2, limn+11n+(1)n1n2,limn+1 + 210n5 + 310n , limn+3n+ (2)n3n+1+ 2n+1 , limn+log2n + 32 log10n + 15 .3. n.limn+(3n24n + 5), limn+(n24n5+ 1), limn+_(1 n)5+ n4_,limn+3n25n5n2+ 2n 6 , limn+2n5+ 4n23n7+ n310 , limn+3n2+ 4n2n 1,limn+_2n 33n + 7_4, limn+_n2+ n + 13n + 1_3, limn+_n2+ n + 13n + 1_4,limn+_n2n + 1 n 1_, limn+_n(n + 1)n + 44n34n2+ 1_.4. , , .limn+(1 + 2 + 22+ + 2n), limn+_1 2 + 22+ + (1)n2n_,limn+_1 1 + 1 1 + + (1)n_, limn+_1 +12+122+ +12n_,limn+_2737+2838+ +2n+63n+6_, limn+_2n3n+2n+13n+1+ +22n32n_,limn+1 + 2 + + 2n1 + 3 + + 3n , limn+1 3 + + (1)n3n1 2 + + (1)n2n .5. ;limn+_(1)n1+10n3_, limn+1 + (1)n1nn, limn+_2n+ (1)n1n_,limn+(1)n1nn + 1 , limn+1(1)n1+1log3 n, limn+1(1)n1n+1n2.(: .)496. xn ,= 1 n x ,= 1. xn x xn1+xnx1+x .(: ,yn=xn1+xn y=x1+x .)7. (xn), (yn) (xn + yn) . (xn), (yn) (xnyn) .8. (xn + yn) (xn), (yn) , , , . (xnyn) (xn), (yn) , , , .9. (xn), (yn) xn 0, yn + (xnyn) .10. [xn[ 0, xn 0.(: n0 .)11. ;limn+_n 1n_ = limn+_1n+ +1n. .n_ = 0 + + 0. .n= 0.limn+_1 +1n_n= limn+_1 +1n_

_1 +1n_. .n= 11. .n= 1.12. : s. , , 12 . , ( ) , 48. , (i) .(ii) .13. vkmhr. , d km, dv hr. ,, :. .50. .1. 2.2,2.4 .2. limn+xn .(i)1 < xn n2+3nn2+1 n.(ii)log10 n22 log10 n+4< xn 00 , x = 0, x < 0(: [a] a < [a] + 1.) [nx] [ny] ___+, x > y0, x = y, x < y5. xn + (yn) . xn + yn +.(: l (yn). . . . .) xn (yn) . xn + yn .6. xn 0 (yn) . xnyn 0.(: [yn[ M n, M[xn[ xnyn M[xn[ n.)7. xn + (yn) . xnyn + ,.xn + (yn) . xnyn +,.8. .22n+(1)n1n 0,_12+(1)n14_n 0.9. 1n2+ 1+1n2+ 2+ +1n2+ n 1+1n2+ n 1.(: 1n2+n 1n2+k 1n2+1 k1 k n.) 1n2+1+1n2+2++1n2+n1+1n2+n0. 0 1. 11.5110. Fibonnaci (xn), x1=x2=1 xn+2=xn+1 + xn(n 1). xn n2 n. (xn);11. 0 a < 1 (xn) : [xn+1[ a[xn[ n. xn 0.(: [x2[ a[x1[, [x3[ a2[x1[, [x4[ a3[x1[ .)a>1 (xn) : xn+1 axn n x1>0. xn +.12. , (xn)[a, b] xn x, x [a, b]. x (xn), (a, b);(: (0, 2) (1n)(2 1n).) x (xn) (a, b);13. 2.5 2.6.14. , . n5+ 4n3< 100 n ; n735n6+ n347n < 84 n;32 1 n2n=12. e _(1 +1n)n_. e = limn+_1 +1n_n. e, , .e , , . - , a0+a1x+ +aNxN= 0 . : mn( m, n) , , m+(n)x = 0 . . , 2 (2) + 1x2= 0. - . , e . , 10, e . e y> 0 log y ln y logey. 1.10 1.11. 2.9. (1) log(yz) = log y + log z y, z> 0.(2) logyz= log y log z y, z> 0.55(3) log(yz) = z log y y> 0 z.(4) log 1 = 0 log e = 1.(5) 0 < y< z, log y< log z. 2.10. a > 0, a ,= 1. logay= log ylog a y> 0.. . 1. , 2, . , 1, P4, P8, P16,. . . 4, 8, 16,. . . . , P2n 2n. P2n P2n+1 : P2n+1 2n P2n 2n P2n , 2n+ 2n= 2n+1. pn P2n , p2=42. pn+1 pn . pn+1=2pn2 +4 pn24n ., , Q4, Q8, Q16,. . . 4, 8, 16,. . . . , Q2n 2n, P2n . qnQ2n , , , q2=8 qn pn qn=pn1 pn24n+1., qn+1=4qn2 +4 +qn24n. (pn) (qn) . , pn+1=2pn2+4pn24n>2pn2+4= pnqn+1=4qn2+4+qn24npn1=pn . ,, :p2< p3 0 y> 1. y> 1 (1, +), , (1, +) (1, +). (, 1) (, 1).(3) y= e2x (, +).ye2x=yx . y 0e2x=y y>0 x = 12 log y. y> 0, (0, +).65(4)y=1 +1x . x1+1x 0 , , x 1 x > 0. (, 1] (0, +). y 1 +1x= y x . ,1 +1x= y ,y< 0. y 0,1 +1x= y (y21)x = 1. y= 1, ,y 0 y ,= 1,x=1y21 . , 1y21 1 1y21> 0. , y 0, y ,= 1 . , [0, 1) (1, +). (0, +) . y 1 +1x= y x (0, +). , y< 0 y= 1, x =1y21 (0, +), 1y21>0. y2> 1, y> 1. (0, +) (1, +)., y 1 +1x=y x (, 1], (, 1] [0, 1)..1. .y=23x 4, y= x24x + 3, y=2x 1x + 4, y=x21x2+ 1 , y=x2+ 1x21 ,y= 2x, y= log10x + 4, y= e2x2ex+ 3, y=ex+ 1ex1y=xx 1 .2. .(i)y= x24x + 3 (, 1], (1, 3], (3, +), (, 2], [2, +).(ii)y=2x1x+4 (, 4), (4, +).(iii)y=x2+1x21 (, 1), (1, 1), (1, +).(iv)y=ex+1ex1 (, 0), (0, +).(v)y=xx1 [0, 1), (1, +).3.3 . y -x . , , :y= x2, y= sin x, xy= 2, y2x3= 0. y x ; y x . 66 y x; xy=2y y y x. ; yx y=2x . y x. , . y2 x3=0 y, . x0 y : y=x3=x32 y= x3= x32 . y2x3= 0 ,y= x32, y= x32 [0, +). . , : . : . y2 x3=0 () , , y= x32..1. . - x y; ; ; .x + y= 1, x22yx + 1 = 0,y xy + x= 2, x3+ y3= 0, x2y2= 0,(xy)2= 1, e(x1)y2= x, y42xy2+ x2= 1, sin(x + y) = 1.3.4 .3.4.1 ,,. y=f(x), , . , , 0 . x y= f(x) y ( x y). x- y-. ,67 x y =f(x) (x, y) = (x, f(x)). , x y=f(x), (x, y) = (x, f(x)) . (x, y) = (x, f(x)) . f= (x, f(x)) : x f., . x, , . (x1, f(x1)), (x2, f(x2)),. . ., (xn, f(xn)) n x . , . 3.1: . y =f(x), . y =f(x). (x, f(x)) . x- x y- y= f(x) . , y= f(x) x- y-.. y= ax + b,a, b , (, +). (0, b). (x, y)y= ax+b , , y b = a(x0). (0, b)68 3.2: . (x, y) ybx0= a. , (x, y) l (0, b) a. . (x, y) l ybx0(0, b) (x, y) a, ybx0=a, y=ax + b, , (x, y) . ,, l.a > 0, l l 3.3: y= ax + b. : a > 0 a < 0. , a < 0, l l . a = 0, l l .(, a ,=0) l y- y-, (, +). ( , y=3x 13.2.) (,a = 0) l , y- b, b.y=f(x)Ix1, x2Ix10, , a < 0. a = 0, (, +), x y b. y= f(x) A u f(x) u x A. u A. y= f(x) A l f(x) l x A l A. ,y=f(x) A A, u l l f(x) u x A.u y=f(x)A, u

u A. , l y=f(x)A, l

l A. u y= f(x) A A y= u. , l y= f(x) A A y=l. , , u l , ,70 y= f(x) A, A y= u y= l., , u y= f(x) A A (, u] l y= f(x) A A [l, +). , u l y=f(x)A, , A [l, u].. y= x2 (, +).y =x2 [0, +) (, 0]. [0, +)(, 0] . y x2= y x [0, +). : y< 0, ,y 0, x = y[0, +). [0, +) [0, +)., x2= y x , y< 0, x = y(, 0],y 0. (, 0] [0, +). y= x2 (, +) 0 3.5: y= x2. [0, +). ,y= x2 (, +) (, 0], [0, +), [0, +) (, u]. [0, +) . (0, 0) (0, 0), (12,14), (1, 1), (2, 4). x- [0, +) 71 y- [0, +), [0, +). . , x , (x, x2), y= x2, ., (, 0] . , (0, 0) (0, 0), (2, 4), (1, 1), (12,14). x- (, 0] y- (, 0], [0, +). . , x , (x, x2), y= x2, . y= x2 () . . y= f(x) f(x) = f(x) x . (a, b) , b = f(a), b=f(a), (a, b) . (a, b) (a, b) y-. , , , y- . y= f(x) y-. 3.6: . y= x2 , y-.. y= x3 (, +). (, +). (2, 8), (1, 1), (12, 18), (0, 0), (12,18), (1, 1), (2, 8).72 3.7: y= x3. y= x3, x3= y x. y 0 , x =3y, y< 0 , x = 3y .,y (, +).y=x3 (, +), (, +). x- , (, +), y- , (, +). . , x , (x, x3) , x , (x, x3) . y= x3 .y= f(x) f(x) = f(x) x . (a, b) y =f(x), b =f(a), b=f(a), (a, b) y=f(x). (a, b)(a, b) (0, 0). , , (0, 0) . y= f(x) (0, 0).y=x3 , (0, 0).. y=1x (, 0) (0, +).y=1x (, 0), (0, +). 73 . 1x= y x. y 0, (0, +), y> 0, x =1y(0, +). (0, +) (0, +)., 1x= yx (, 0), y 0, x =1y(, 0),y< 0. (, 0) (, 0).y=1x (0, +), , 0 (0, +). (, 0) , ,0 (, 0). (0, +) . 3.8: y=1x . (12, 2), (1, 1), (2,12). x- (0, +) y- (0, +), (0, +). , x , (x,1x) ( y=1x) , x , (x,1x) . , y- x-., (, 0). (2, 12), (1, 1), (12, 2). x- (, 0) y- (, 0), (, 0). , x , (x,1x) , x , (x,1x) . , x- y-. y=1x , , 74 .y=1x , (0, 0): (0, 0). . (a, b) , b =1a , a =1b , (b, a) . (a, b) (b, a) y=x, . , , . , , . . ,- . y =1x . y=1x : (, 0) (0, +). 0 . . . . , -. - -. - . 6.7. , . , . 3.9: y= [x].. y= [x], x, (, +) .y =[x] [k, k+ 1), k : y=k x[k, k + 1). 75[k, k +1) . , y= [x] [k, k+1), ., y= [x] ( (, +) ) - .3.4.2 .. . , , . y=f(x) , , . y= f(x) ,, . . , x- y- . . . , [0, +). y- (0, 0).. y=f(x). y= f(x).(1) (x, f(x)) y= f(x) x- (x, f(x)) y= f(x). : y= f(x) x- y= f(x). 3.10: y= f(x) y= f(x).76(2) (x, f(x)) = (x

, f(x

)) y= f(x) y- (x, f(x)) y= f(x). : y= f(x) y- y= f(x). 3.11: y= f(x) + y= f(x ).(3) . (a, b) (a, b + ). (x, f(x) +) y= f(x) + (x, f(x)) y= f(x). : y= f(x) + y= f(x).(4) . (a, b) (a + , b). (x +, f(x)) = (x

, f(x

)) y= f(x ) (x, f(x)) y= f(x). : y= f(x ) y= f(x).(5) . 3.12: y= f(x) y= f(x).(a, b) (a, b). (x, f(x)) y= f(x) (x, f(x)) y= f(x). :77 y= f(x) y= f(x).(6) . (a, b) (a, b). (x, f(x)) =_x

, f_x

__ y= f_x_ (x, f(x)) y= f(x). : y= f(x) y= f(x)..1. .y= [x[, y= [x[x, y=___1 , x > 00 , x = 01 , x < 0y= x [x],y= (1)[x], y= x(1)[x], y= (1)[1x], y= x(1)[1x]. ; ; ; . ; ;2. y=x2 y=x21 .3. y= x2, :y= 3x2, y= x24, y= (x + 4)2, y= (3x + 4)2, y= 4 (3x + 4)2. , , . ,,( ) (, +).4. a ,=0 b, c. y=x2, y= ax2+ bx + c.(: ax2+ bx + c = a(x +b2a)2+4acb24a.) (, +);5. y=1x , :y=1x+ 2, y=1x + 2 , y=13x + 2 , y=3x + 2 .6. a, b, c, d c ,= 0. y=1x , y=ax + bcx + d .78(: ax+bcx+d=ac(cx+d)+badccx+d=bcadc21x+dc+ac .)y=ax+bcx+d . y=2x+33x1 . , , .7. > 0. y= f(x) , y= f(x), y=f(x), y=f(x) + , y=f(x ), y=f(x)y= f(x);8. y= x23x+2 , , y= [x23x+2[ y= [x[23[x[ + 2. ;:(i) y= f(x) y= [f(x)[.(ii) y= f(x) y= f([x[).3.5 . y =f(x) , , , y x , ,x y. , , y y=f(x) x. , yf(x)=y x . , f, f1 x = f1(y) .:y= f(x) x = f1(y) x y f. . f(x)=y x. y x, y= f(x1) =f(x2) x1= x2, -- . -- ,, .y =f(x) x=f1(y) : (x, y) y=f(x) (y, x) x = f1(y). , (x, y) (y, x) , y= x,: .79 3.13: x = f1(y). x=f1(y) y =f(x), ., x = f1(y) y- x-. ; , x=f1(y) y=f(x), x- y- . x y, y=f1(x), x=f1(y)y=f(x). ,, : x- y- . : , ,., y= f(x) y1, y2 x = f1(y) y1< y2 . x1= f1(y1),x2= f1(y2) x = f1(y) f(x1) = y1, f(x2) = y2.x1= x2 ,,, y1= f(x1) = f(x2) = y2 . x1> x2 , , y= f(x) , y1= f(x1) > f(x2) = y2 . , x1< x2 , , f1(y1) < f1(y2). y= f(x) x = f1(y). y= f(x) , x = f1(y), , . . , 80 ., y =f(x) , x=f1(y), , . 3.14: x =3y .. y= x3.y=x3 , (, +), (, +). , x=_3y , y 03y , y< 0 (, +) (, +). . , ., , y=_3x, x 03x, x < 0y=f(x) --, . ,I y=f(x) , x I, y=f(x) --: x1, x2I f(x1) = f(x2) x1= x2 . y= f(x) I( ) ( ) I. y=f(x) --, , . , , I .. y= x2 (, +).y= x2 -- y> 0 x2= y :x = y x = y ., [0, +) y=x2 ,81 3.15: x = y ., -- [0, +). x = y , [0, +) [0, +) . ( ) , ., (, 0]y= x2 , 3.16: x = y ., -- [0, +). x= y , [0, +) (, 0] . ( ) , .3.3 y=x2(, +) [0, +)x= y[0, +) (, +)..1. y=13x+1 .82 , . , , .2. a, b, c, d c ,= 0 y=ax+bcx+d 6 . , . y=2x+33x1 .3. y= x2+ 4x + 1. , . ; , . , . y= x2+ 4x + 1 ;3.6 . 3.17: y= xn. y= a0 + a1x + + aNxN.83aN ,= 0, N . , , (, +). y= xn (,) n. y=x3, n , y=xn , (, +). , (0, 0), (1, 1), (0, 0), (1, 1) ., y= x2, n , y= xn , [0, +) [0, +) (, 0] [0, +). , y-, (1, 1), (0, 0), (1, 1) (0, 0) (0, 0) . , y=xn(0, 0), (1, 1). (0, 1) y= x , y= x2, y= x3,. . . (1, +) . , , x (0, 1) xx-: x>x2>x3>. x(1, +) x : x < x2< x3 0, (0, +),a < 0. a. y=xa a . ,, , a, y= xa x, y= xa (, 0).87,, a , , y= xa . y= xa, xa= y x. y< 0, . y= 0, x = 0 , a>0, , a0, x = y1a . ,a > 0, y=xa [0, +), a 0, (0, +), a0, y=xa (0, 0),(1, 1) x- [0, +) y- [0, +). (0, 0) . , a 0 < 0..1. ;y= x0, y= x3, y= x3, y= x46, y= x46,y= x64, y= x64, y= x2, y= x2. .882. y= x2, y= x2, .y= (2x 3)2, y= 2 (2 3x)2, y= (1 x)2, y= 3 + (2x + 1)2. .3.9 . a > 0 y= ax (, +) a.a = 1, , y= 1x= 1, 1. 3.21: y= ax x = logaya > 1. 3.22: y= ax x = logay0 < a < 1. a>10 1,,0 < a < 1.89 y=ax (0, 1), (1, a). a>1, x- (, +) y- (0, +) . x- . , 0 0 f(x T) = f(x)90 x. , , , x y=f(x), x T . T y= f(x)..(1) y= cos x y= sin x 2, cos(x 2) = cos x sin(x 2) = sin x.(2) y= tan x y= cot x , tan(x ) =tan x cot(x ) = cot x. y= f(x) T. f(xT) = f(x) y= f(xT) y= f(x) , , . , , y= f(x+T) y= f(x). T f f. . y= f(x) T. a y= f(x) [a, a + T].k y=f(x) [a + kT, a + (k+ 1)T] kT [a, a + T]. , y= f(x) T. [a, a+T] T. .1. y= cos x. 3.23: y= cos x. (, +) [1, 1]. 2 . [, 0] [0, ]. [1, 1]. [, ]: (, 1) (0, 1) (0, 1) (, 1) (2, 0), (2, 0).912. y= sin x. (, +) [1, 1]. 2 . [2,2] [2,32]. [1, 1]. [2,32]: (2, 1)(2, 1) (2, 1)(32, 1) (0, 0), (, 0). 3.24: y= sin x.3. y= tan x. (2 +k,2 +k)(k Z). (, +). . (2,2) (, +). (2,2): (0, 0) x = 2 x =2.4. y= cot x. (k, (k + 1))(k Z). (, +). . (0, ) (, +). (0, ): (2, 0) x=0 x = . y= cos x y= sin x (, +) 1 1.92 3.25: y= tan x y= cot x.3.10.2 . , - . .y= cos x [0, ] [1, 1]., x= arccos y, [1, 1] [0, ].y= sin x [2,2] [1, 1]. x = arcsin y, [1, 1] [2,2].y= tan x (2,2)(, +). x= arctan y, (, +) (2,2).y= cot x (0, ) (, +). x = arccot y, (, +) (0, )., , , , x y,:1. - y= arccos x. [1, 1] [0, ]. [1, 1] (1, ) (1, 0) (0,2).2. - y= arcsin x. [1, 1] [2,2]. [1, 1] (1, 2)(1,2) (0, 0).93 3.26: y= arccos x y= arcsin x.3. - y= arctan x. (, +) (2,2). (, +) y= 2 y=2 (0, 0). 3.27: y= arctan x y= arccot x.4. - y= arccot x. (, +) (0, ). (, +) y= y= 0 (0,2)..1. .y= cos(2x), y= tan_x2 1_, y= 1 + 2 sin(1 3x), y= cot(1 x),x = 2 arccos(2y + 1), x =2+ arctan(1 y), x = arctan_y + 12_.942. 5 1.4 y= a cos x + b sin x. y= cos x + sin x, y=3 cos x + sin x, y=3 cos x sin x.3. .y=sin x, y=11 + sin x , y= log(sin x), y= arcsinxx 1 .4. y= arcsin x y= arctan x; ;(: x = sin y x = tan y.)5. y= arccos(cos x),y= arcsin(sin x), y= arctan(tan x), y=arccot(cot x).6. y= xsin x [0, +). , x x sin x x, y=xsin x y= xy= x sin x = 1, x =2 +k2 (k = 0, 1, 2, . . . ), y =x sin x y =x sin x= 1, x=32+ k2 (k=0, 1, 2, . . . ), y =x sin x y = x. y =x sin x x-; y= x sin x [0, +) , y= x sin x , (, 0].7. y= sin1x(0, +). y= sin1x y= 1 y= 1. sin1x=1 sin1x= 1 (0, +). (0, +) - 0 y= sin1x . sin1x= 1 y= sin1x y= 1sin1x= 1y= 1. y= sin1x x-; y= sin1x (0, +) , y= sin1x , (, 0).8. , .y= x2sin x, y=xsin x, y=1x sin 1x .953.11 .3.11.1 . x cosh x =ex+ ex2, sinh x =exex2,tanh x =exexex+ ex, coth x =ex+ exexex . , , - x,. 3.1. (1)(cosh x)2(sinh x)2= 1.(2) tanh x =sinh xcosh x ,coth x =cosh xsinh x .(3) cosh(x) = cosh x,sinh(x) = sinh x,tanh(x) = tanh x,coth(x) = coth x.(4) cosh(x + y) = cosh x cosh y + sinh x sinh y, sinh(x + y) = sinh xcosh y + cosh xsinh y.(5) cosh x cosh y= 2 sinhxy2sinhx+y2,sinh x sinh y= 2 sinhxy2coshx+y2.(6)(i)1 cosh x < cosh x

,0 x < x

(ii)1 cosh x < cosh x

,x

< x 0.(7) sinh x < sinh x

,x < x

. 3.1 . 3.1 1.12. cosh x=ex+ex2 sinh x =exex2. cos x =eix+ eix2, sin x =eixeix2i eix= cos x +i sin x eix= cos x i sin x. . y= cosh x =ex+ ex2 (, +). . 3.1 y= cosh x [0, +) (, 0]. y= cosh x ex+ex2= y x e2x2yex+1 = 0. t = ex, t22yt+1 = 0 = 4y24. .(i) y2< 1. t22yt + 1 = 0 .(ii) y2=1. y= 1, t2 2yt + 1=0 ex=t = 1, ,. y= 1,t22yt + 1 = 0 ex= t = 1 x = 0 cosh x = 1.(iii)y2>1. t2 2yt + 1=0 () 2y1. y< 1, , . ,y> 1, t22yt + 1 = 0 , . 1, 1 0 1. 96 t=y y21 0 0 , 0 < [x [ < x , [f(x) [ < ..(1) y=3x2x2x1. ( )>0 >0, 0 < [x1[ < x , [3x2x2x15[ < . ,0 < [x 1[ < , x , 1 x x. >0, 0< [x 1[ 0 > 0 , 0 < [x [ < x , f(x) > M.. y =1(x1)2 . ( ) M>0 >0, 00, 0< [x 1[ M. 1(x1)2>M (x 1)2 0 , 0 < [x[ < x , f(x) > M. : M> 0 > 0 0 < [x [ < x f(x)>M. : M>0 >0 f(x)>M x 0 < [x [ < . limxf(x) = + y=f(x) + + +x ..(1) a >0. y =[x [a=1|x|a (, ) (, +), . , a=1, a=2( -) a=12 , , x ,=, y= [x [a . , limx[x [a= +. , . M> 0 > 0 [x[a> M x(x ,=) 0< [x [ M 0 < [x[ < M1a , , = M1a , x 0 < [x[ < 0 < [x[ < M1a, , [x[a> M.:limx[x [a= + (a > 0).: limx1|x|= +,limx1(x)2= +,limx1|x|= +.(2) y =x+2(x+1)2 (, 1) (1, +) 106 1. x 1,= 1, y =x+2(x+1)2 ,, limx1x+2(x+1)2= +. . M> 0 > 0 x+2(x+1)2> M x ( x ,= 1) 0 < [x + 1[ < . x ,= 1, x+2(x+1)2>M (x + 1)2 0 >0 f(x)< M x 0 < [x [ < . limxf(x) = y=f(x) x . f(x) < M f(x) >M , , limx f(x)= limx_ f(x)_=+. . 1 1 ..(1) limx1_x+2(x+1)2_ = .(2) limx_[x [a_ = (a > 0). 2. y= f(x) (a, )(, b) . , ,,. ,, x , < > . . . ,, .2 : = . > 0 > 0 [f(x) [ 0 > 0 [f(x) [ 0 >0 f(x)>M x < x < + . limx+f(x) = + y=f(x) + + +x ., M> 0 > 0 f(x) > M x < x < . limxf(x) = + y=f(x) + + +x .2 : = . M>0 >0 f(x)< M x < x < + . limx+f(x) = y=f(x) x ., M> 0 > 0 f(x) < M x < x < . limxf(x) = y=f(x) x . 2 limx+f(x) limxf(x) - y= f(x) ,. limx f(x) limx+f(x) limxf(x). 4.1.4.1. y=f(x) () (a, ) (, b) .108 limx f(x), limx+f(x) limxf(x) :limxf(x) = limx+f(x) = limxf(x)., limx+f(x) limxf(x) , limx f(x) .. . + .limx f(x)=. >0, >0 [f(x) [ 0 (, ). 1x< M (xM x x > N. limx+f(x) = + y=f(x) + + +x +.4 : = ., M>0 N>0 f(x)< M x x > N. limx+f(x) = y=f(x) x +..(1) y=x+1x+3 (, 3)(3, +), +. , , x , y=x+1x+3 1. ,,limx+x+1x+3= 1. . > 0 N> 0 x+1x+3 1 N. x+1x+3 1< 2|x+3|< [x + 3[ >2

111x< 3 2

x> 3 +2

(: x ) x > 3 +2

. N> 0 3 +2

. 3 +2

>0, , N( N 3 +2

) x > 3 +2

,, x+1x+3 1 < . limx+x+1x+3= 1.(2) y=x 7x (, 0) (0, +), +. x, y= x7x . limx+_x7x_ = + . M> 0 N> 0 x 7x> M x ( x ,= 0) x > N. x 7x>M x2Mx7x>0 x(x2 Mx 7)>0MM2+282

M+M2+282. N=M+M2+282>0, x( x ,=0) x>Nx >M+M2+282,, x 7x> M. limx+_x 7x_ = +.(3) y= c. > 0 N> 0 [yc[ = [cc[ = 0 N. :limx+c = c.(4) a >0. y =xa [0, +). limx+xa= +. M> 0 N> 0 xa> M x x > N.x > 0, xa> M x > M1a ,, N= M1a> 0, x > N x > M1a,, xa> M. :limx+xa= + (a > 0).: limx+x = +,limx+x2= +,limx+x = +,limx+5x = +.(5) a>0. y =xa (0, +). limx+xa= 0. > 0 N> 0 [xa0[ N.x > 0, [xa0[ 1a ,, N= 1a> 0, x x > Nx > 1a, , [xa0[ < . :limx+xa= 0 (a > 0).: limx+1x= 0,limx+1x= 0,limx+1x3x= 0.112 5. y= f(x) (, b), . x f(x).5 : = . > 0 N> 0 [f(x) [ 0 N>0 f(x)>M x x < N. limxf(x) = + y=f(x) + + +x .5 : = ., M>0 N>0 f(x)< M x x < N. limxf(x) = y=f(x) x ..(1) y= c. > 0 - N> 0 [y c[ = [c c[ = 0 0 N> 0 3x7+x (1) 0, x ( x ,= 7) x < N x < 7 10

,,3x7+x (1) < . limx3x7+x= 1.113.1. ;limx11x22x, limx0x2, limx1log(x 1),limx1+1 x2, limx+1log(3 x) , limx4 + 3x x2.2. ; . x.limx2(x2+ 1), limx1x + 2x + 1 , limx11(x 1)2 , limx11x21, limx1+1x21,limx+1x3 , limx+1x + 5, limx1x2+ 5, limxx23 .3. .limx13x = 3, limx2_x2 7_ = 6, limx11x= 1, limx1x2= 1, limx2x + 1x 1= 3,limx1log x = 0, limx0ex= 1, limx2x =2 , limx1xx 1 = +,limx3_x + 1x + 3_2= +, limx21 x(x 2)2= , limx1+(3x 2) = 5, limx2+x2= 4,limx0xx + 1= 0, limx1+3x 1= , limx2+x2 x= , limx1x + 21 x= +,limx0+log x = , limx+ex= +, limx+log x = +, limxx 32x + 1=12 ,limxex= 0, limx(3x2+ x) = +, limx+2 x2x + 1= , limx2 x2x + 1= +.4. 1. . x 1 , , ;y=_2x + 3, x > 11 2x, x < 1y=_2x 1, x > 1xx1 , x < 1y=_x , x 1x2, x < 1y=_2xx1 , x > 1(x 1)2, x < 1114 4.1: limx f(x) = .4.2 . . y= f(x) (a, ) (, b) x (a, ) (, b). x x- f(x)y-, , (x, f(x)) . f(x) (x, f(x)) . ,f(x) x ,=, , limx f(x)= (x, f(x)) (, ) x=. , f(x) x ,= ,,limx f(x) = + (x, f(x)) x = .: limx f(x) = (x, f(x)) x = . 4.2: limx f(x) = +. . , limx f(x) = , x = y= f(x). 115 4.3: limx f(x) = . .limxf(x). limx+f(x) (x, f(x)) x= limxf(x) (x, f(x)) x = .x= y=f(x) limxf(x) = ., , limx+f(x) limxf(x) , limx f(x). 4.4: limxf(x) = limx+f(x) limxf(x) ,= limx+f(x). y=f(x) (a, +) x x-. f(x) y- (x, y) = (x, f(x)) . , f(x) , , limx+f(x)= (x, f(x)) y =. y= (+) y=f(x). :limx+f(x)=+ (x, f(x)) . : limx+f(x)= (x, f(x)) . , y =f(x) (, b), limxf(x) = (x, f(x)) y=. y=( ) y=f(x). : limxf(x)=+ (x, f(x)) . :116 4.5: limx+f(x) = ,limx+f(x) = + limx+f(x) = . 4.6: limxf(x) = ,limxf(x) = + limxf(x) = .limxf(x)= (x, f(x)) . 4.7: < f(x) < + x < x < < x < + . limx f(x) = y= f(x) . > 0 > 0 x ( , ) (, +) f(x) (x, f(x)), + , , 117 ( , ) (, + ) y= y= + . . , limx+f(x)= M>0 N> 0 y= f(x) (N, +) y= M. . l y= x + () + y= f(x) y= f(x) (a, +) limx+_f(x) x _ = 0. (x, f(x)) y= f(x) (x, x + ) l . : l +. 4.8: + . () y= f(x) (, b). l y= x + limx_f(x) x _ = 0. 4.3.2 , , . y=f(x) . , 0 ( = 0)..1. y =f(x) =_x21 , x < 21x , x 2 limx2f(x),limx0f(x) limxf(x) ; y= f(x) =_x2, x 21x , x < 2, x ,= 01182. :y= ax+b, y= xn(n ), y= xa, y= ax, y= logax, y= [x], y= cos x,y= sin x, y= tan x, y= cot x, y= arccos x, y= arcsin x, y= arctan x,y=arc cot x, y= cosh x, y= sinh x, y=arc cosh x, y=arc sinh x. , ( ) . ;4.3 .4.3.1 . y=f(x) , , f(x)x ,=. ,, y= g(x) (a, )(, b), g(x) = f(x) x (a, )(, b). (a, ), (, b) , , x . y= f(x) x , y= g(x) ., y= g(x) y= f(x) y=f(x), (a, ) (, b), a b . 4.9: limx f(x) = limx g(x) limx+f(x) = limx+g(x)., g(x)=f(x) x(, b) y=f(x) x +, y=g(x) ., g(x)=f(x) x(a, ) y=f(x) x , y=g(x) . , g(x)=f(x) x (a, +) y= f(x) x +, y= g(x) . x .:1194.2. y=f(x) y=g(x) (a, ) (, b)(, b)(a, ) (a, +) (, b). ,, x x + x x + x , .. y=f(x) y=g(x) (a, ) (, b) limx f(x) = . . > 0. limx f(x) = ,

> 0 [f(x) [ 0, > 0 [f(x) [ < x y= f(x) 0 < [x[ < . [(f(x))()[ =[ f(x)[ = [f(x) [. [(f(x)) ()[ 0, > 0 f(x) > M x y=f(x) 0< [x [0,

>0[f(x) [ M2 x y=f(x) 0< [x [ 0 g(x) >M2 x y= g(x) 0 < [x [ <

. =min

,

.

, f(x) >M2 g(x) >M2 x y=f(x) y=g(x) 0< [x [ M2+M2= M x y= f(x) +g(x) 0 < [x[ < . limx(f(x) + g(x)) = + = (+) + (+). limx f(x) = + limx g(x) = . M> 0,

> 0 f(x)>M + 1 x y=f(x) 0< [x [ 0 [g(x) [ < 1 x y= g(x) 0 < [x[ <

.= min

,

.

, f(x) > M +1 [g(x) [ < 1 x y= f(x) y= g(x) 0 < [x[ < . [g(x)[ < 1 g(x)> 1, , f(x) + g(x)>(M + 1) + ( 1)=M x y= f(x) + g(x) 0 < [x [ < . limx(f(x) + g(x)) = + = (+) + . ..(1) limx0 1 = 1 limx01|x|= +. limx0_1 +1|x|_ = 1 +(+) = +.(2) limx0+x = 0,limx0+ 1 = 1,limx0+1x= + limx0+1x= +. limx0+_x + 1 +1x+1x_ = 0 + 1 + (+) + (+) = +.(3)limx+(1)= 1, limx+1x=0 limx+1x=0. limx+_ 1 +1x+1x_=1 + 0 + 0 = 1. ..(1) limx0+(1x+ 3) +, limx0+(1x)= limx0+_(1x+ 3) +(1x)_ = limx0+ 3 = 3., 3.(2) limx+ 2x = +,limx+(x) = limx+_2x + (x)_ = limx+x = +.(3) limx0(1x) = +,limx02x= limx0(1x+2x) = limx01x= .(4) limx0(2x2) = . ( ) limx0(2x2+1x) =+. , limx0_(2x2+1x) + (2x2)_ = limx01x . y= f(x) y= g(x), y= f(x)g(x), f(x) g(x) = f(x) + (g(x)). :121. limf(x), limg(x) limf(x) limg(x) , lim(f(x) g(x)) lim(f(x) g(x)) = limf(x) limg(x)..(1) limx+ 1 = 1 limx+1x= 0. limx+_1 1x_ = 1 0 = 1.(2) limx1x = 1 limx11|x1|= +. limx1_x 1|x1|_ = 1 (+) = .(3) limx0 1=1, limx01x= limx01x=+. limx0_1 +1x 1x_=1 + () (+) = . y=f(x) y=g(x) y=f(x)g(x) x .. limf(x), limg(x) limf(x) limg(x) , lim(f(x)g(x)) limf(x)g(x) = limf(x) limg(x).. limx f(x)= limx g(x)=. >0,

>0 [f(x) [ 0 f(x)>M x y=f(x) 0< [x [ 0 g(x) >M x y= g(x) 0 < [x [ <

. =min

,

,

. f(x)>M g(x)>M x y=f(x) y=g(x) 0< [x [ MM=M x y=f(x)g(x) 0< [x [ 0. M> 0,

> 0 f(x) >2M x y= f(x) 0 < [x[ <

> 0 [g(x) [0. >0, >0 [f(x)[ < min2

2,2 x y= f(x) 0 < [x[ < . [f(x) [ 1

x y=f(x) 0< [x [ 0 x (a, )(, b). M> 0,

> 0 [f(x)[ = [f(x)0[ 0 x. ,

, 1f(x)=1|f(x)|> M x 0 < [x [ < . limx1f(x)= +.(2) (1).. limx+(1x 1x2) = 0. 1x 1x2> 0 x > 1. limx+11x1x2= +. y =f(x) y =g(x), y =f(x)g(x), . . limf(x), limg(x) limf(x)limg(x) , limf(x)g(x)lim f(x)g(x)= limf(x)limg(x) .125.(1) limx1(x2+ x) = 2 limx1(x 2) = 1. limx1x2+xx2=21= 2.(2) limx+x = +, limx+(1 1x +1x2) = 1 0 +0 = 1 limx+(1 +21x) = 1 +20 = 1. limx+x2x+1x+2= limx+x11x+1x21+2x= (+)11= +. limx+x2x+1x+2 . 00++ ..(1) limx0+(2x) = 0,limx0+x = 0 limx0+2xx= limx0+(2) = 2. 2 .(2) limx0+x = 0,limx0+x2= 0 limx0+xx2= limx0+1x= +.(3) limx0+x2= 0,limx0+x = 0 limx0+x2x= limx0+x = 0.(4) limx+ 5x = +,limx+x = + limx+5xx= limx+ 5 = 5. 5.(5) limx+x2= +,limx+x = + limx+x2x= limx+x = +.(6) limx+x = +,limx+x2= + limx+xx2= limx+1x= 0. , , . y =f(x) (a, +) +y=x + . . limx+(f(x) x ) =0 limx+f(x)xx=0, , = limx+f(x)x. , , , limx+f(x)x , +. , , , . , , = limx+(f(x) x), . , limx+(f(x) x) , +. , , .. y= x +1x (, 0) (0, +). : = limx+1x(x+1x) = 1 = limx+(x+1x1x) = 0. + y=1x + 0=x. : =limx1x(x +1x) =1= limx(x +1x 1x) = 0. y= 1x + 0 = x. . limf(x), lim[f(x)[ lim[f(x)[ = [ limf(x)[.. limx f(x) = . > 0, > 0 [f(x) [ < x y =f(x) 0 0 f(x) >M f(x) < M, , x y=f(x) 0 < [x[ < . [f(x)[ > M x y= f(x) 0 < [x [ < . limx[f(x)[ = + = [ [.126.(1) limx1(x 2) = 1, limx1[x 2[ = [ 1[ = 1.(2) limx0(1x 1x2) = () (+) = , limx0[1x 1x2[ = [ [ = +.(3) . y= f(x) =|x|x (, 0) (0, +). limx0[f(x)[= limx0|x|x= limx0 1=1. , limx0f(x) .4.3.3 . z= g(f(x)) : y= f(x) z= g(y). , , y =f(x) z= g(y). ., limx f(x) = limy g(y) =. , , . x ,= , y= f(x) , , . , , y= f(x) ,= , y= f(x) ,= ., z= g(f(x)) = g(y) . , , x ,= z= g(f(x)) = g(y) . limx g(f(x)) = . , z= g(f(x)), y. , (i) x y= f(x) f(x) y (ii) limx limy. ., . z=g(f(x)) y= f(x) z= g(y). limf(x) = y= f(x) x limy g(y), limg(f(x)) = limy g(y). . :limg(f(x)) =___limy g(y), f(x) f(x) ,= xlimy+g(y), f(x) f(x) > xlimyg(y), f(x) f(x) < xlimy+g(y), f(x) +limyg(y), f(x) . y =f(x) () (a, ) (, b) z =g(x) () (c, ) (, d) limx f(x) =limy g(y) =. f(x) ,= x (a

, ) (, b

), a a

< < b

b. limx g(f(x)) = . >0,

>0 [g(y) [ 0. limx f(x) = +,

> 0 f(x) > M x y=f(x) 0< [x [ 0,

> 0 [f(x) [ nlimxf(x), xn xn< nlimx+f(x), xn +limxf(x), xn . limx f(x) = (xn) xn ,= ,, xn . f(xn) . >0. limx f(x)=, >0 [f(x) [ 0 N aN< 0 N, aN< 0 N aN> 0 N limxp(x) ,limxp(x) = limxaNxN.. limx+(5x3+ x2 4x 12)= , limx(5x3+ x2 4x 12)=+,limx(7x4+ x3x + 5) = +. . r(x) =a0+a1x++aNxNb0+b1x++bMxM aN ,= 0, bM ,= 0, r(x) =aNbMxNMa0aN1xN ++aN1aN1x+1b0bM1xM ++bM1bM1x+1,limx+r(x) =___aNbM (+), N> MaNbM, N= M0, N< M136limxr(x) =___aNbM (+), N M aNbM (), N M aNbM, N= M0, N< M limxr(x) ,limxr(x) = limxaNxNbMxM.. limx+x3x2+2x+42x3+1=12 , limx+x3+3x2+x+42x2x+1= , limx+3x2x+2x45x3+x21=0,limx3x2+2x+42x21=32 , limxx3+5x243x4+1= 0, limxx3x+52x+1= , limx3x4x2+42x+1= . x , .b0 + b1 + + bMM,= 0, limx r(x) =a0+a1++aNNb0+b1++bMM= r(). ,limxr(x) = r().. limx13x2x4+2x34=31214+2134= 1. b0 + b1++ bMM=0, x b0 +b1x + +bMxM. (x )m(m 1) x b0+b1x++bMxM, b0+b1x++bMxM= (x)mq(x), q(x) x ,, q() ,= 0. ,x a0+a1 + +aNN , a0+a1 + +aNN=(x )np(x), n 0 p(x) x , , p() ,=0. , , r(x)=(x )nmp(x)q(x), limxp(x)q(x)=p()q() ,= 0,limxr(x) =_0 , n > mp()q() , n = mlimxr(x) =p()q()(+), mn ,limxr(x) =p()q()(), limx+r(x) =p()q()(+), mn ..(1) limx1x3x2x+1x42x2+1 1 x4 2x2+ 1, , x 1. , , : x42x2+1 = (x21)2= (x1)2(x+1)2. 1 x3x2x + 1 x 1 . ,137: x3x2x + 1 = (x 1)x2(x 1) = (x 1)(x21) = (x 1)2(x + 1). x3x2x+1x42x2+1=(x1)2(x+1)(x1)2(x+1)2=1x+1 x ,=1, 1. , limx1x3x2x+1x42x2+1= limx11x+1=11+1=12 .(2)limx1x3+4x2+x6x3x2x+1 1 x3 x2 x + 1, :x3x2x+1 = (x1)2(x+1). 1 x3+4x2+x6 x1 : x3+4x2+x6 = x3x2+5x25x+6x6 = (x1)x2+(x1)5x+(x1)6 =(x 1)(x2+ 5x + 6). x 1 x2+ 5x + 6 1 . x3+4x2+x6x3x2x+1=(x1)(x2+5x+6)(x1)2(x+1)=1x1x2+5x+6x+1. : limx1+x3+4x2+x6x3x2x+1=(+) 12+51+61+1=+ limx1x3+4x2+x6x3x2x+1= () 12+51+61+1= . , limx1x3+4x2+x6x3x2x+1..1. x .y= x44x3, y=x2+ 1x31 , y=x4x + 13x4+ x2, y=1 x81 + 2x2 , y= 2x + x51 x2.2. x 1.y= x2+ 2x, y=x22x + 1x + 1, y=x3+ 2x22x4x31, y=x4x33x2+ 5x 2x4+ x34x2+ x + 1,y=x + 2x42x3+ 2x22x + 1 , y=x3x2x + 1x53x4+ 6x310x2+ 9x 3 .3. limx1f(x) f(x)_x3x2+2x2x3x2x+1, 0 x < 1x3x2+2x2x3x2x+1, x > 14. limx+f(x) x3+x23x72x3+8x2+x+3 f(x) 5.5. , (i)540. >0 >0[xaa[ 0 0 < [x [ < . [xaa[ 01, a = 00, a < 0 . limx xa 0.)6. x +.y=x2+ 1 x, y=x_x + 1 x_, y= x_x2+ 1 x_,y=3x + 1 3x, y=3x2_3x + 1 3x_,y=x + 1 2x +x 1 , y=x3_x + 1 2x +x 1_.7. .limx0x2+ 1 , limx3x27x + 1x2+ 1, limx+x +x +x, limx31 +1x ,limx051 1x+1x2 , limx1_1 1x 1_23, limx1+x+1x1 37x+1x1+ 12x+1x1+ 75x+1x1+ 3.1408. a, b, c a > 0. A, B a, b, c limx+_ax2+ bx + c Ax B_ = 0. A, B limx+x_ax2+ bx + c Ax B_ =4ac b28aa.9. (xn) a > 0. 4.9, (i) xn + xn> 0 n,xna +.(ii) xn 0 xn> 0 n,xna 0. a < 0; .limn+_n3+ n + 12n21_2, limn+4n5+ n3+ 12n6+ n2+ 1 , limn+_2n4n+ 1_34,limn+5n3+ n + 12n21, limn+52n4n2n3n+ 1 .4.7 . y= ax, a > 0. y= ax (, +). :limxax= a.a > 1. > 0 > 0 [axa[ < x 0 < [x [ < . [axa[ 1. M>0 N>0 ax>M x > N. ax> M x > logaM. N= logaM> 0,M> 1, N= 1 > 0, 0 < M 1, x > N x > logaM, , ax> M. limx+ax= +.0 < a < 1 (1a> 1), limx+ax= limx+1(1a)x=1+= 0.,a = 1, limx+ 1x= limx+ 1 = 1.:limxax=___0 , a > 11 , a = 1+, 0 < a < 1 , . ,, y= x. ,a > 1, limxax= limy+ay= limy+1ay=1+= 0. a = 1 0 < a < 1. . a>0, a ,=1 y= logax(0, +). :limxlogax = loga (> 0).a>1. >0 >0 [ logax loga[ 0) 0 < [x [ < . [ logax loga[ 1. M> 0 N> 0 logax > M x y= logax ( x > 0) x > N. logax > M x > aM,, N= aM> 0, x > N x > aM,, logax > M. limx+ logax = +.0 < a < 1, limx+ logax = limx+(log1ax) = (+) = .:limx0+logax =_, a > 1+, 0 < a < 1142 y=1x . a>1, limx0+ logax= limy+ loga1y= limy+(logay)= limy+ logay=(+) = . 0 < a < 1. a = e, :limxex= e, limxex= 0, limx+ex= +limxlog x = log (> 0), limx0+log x = , limx+log x = +..1. x .y= exe2x+ 2, y=1ex1 , y=e2x+ ex+ 12e2xex+ 2 .2. x + x 0+.y= (log x)2log x, y=1log x , y=1 + 2(log x)22 + log x + (log x)3 .3. .limx01ex1 , limx01(ex1)2 , limx0e2x1ex1,limx0+log(2x)log(3x) , limx11log x , limx11(log x)2 .4. .limx2e1x, limx0e11x, limxe1x, limx+ex+ ex2+ 12exex3+ 2,limx+log(x + 1), limxlog(x2x + 1), limx1log(x3+ 1),limx(log [x[)7(log [x[)4+ 1(log [x[)5+ (log [x[)2+ 1 , limxlogexex2+ 1.5. y= cosh x, y= sinh x, y= tanh x, y= coth x : limx , limx . limx0 coth x.1436. 0 0 ;y= ex, y= e|x|, y=1ex1 , y=1(ex1)2 ,y= log [x[, y=1log [x[ , y=1log [1 + x[ .7. (xn) a > 1. 4.9, (i) xn +,axn +.(ii) xn ,axn 0. .limn+2n, limn+en3+3n1n2+1, limn+en, limn+22n, limn+21nn1+n.8. (xn) a > 1. 4.9, (i) xn + xn> 0 n, logaxn +.(ii) xn 0 xn> 0 n, logaxn . .limn+logn + 12n21 , limn+log n3n2+ 1n2+ 1, limn+log e2n+ 1en+ 1,limn+log(3n2n+ 1), limn+(lognn2+1)2lognn2+1+ 2(lognn2+1)2+ 4 lognn2+1 8 .4.8 . y= sin x.[ sin x[ [x[.. 0 0 =. x y=cos x( x) 0< [x [ 0, limx0+xasin1x= 0.5. limx+ sin x, limx0+ sin1x .6. (xn). xn 0 xn ,=0 n, sin xnxn11cos xnxn212 . .limn+nsin n , limn+nsin n , limn+n2sin n , limn+n2_1 cos n_, limn+cot2nn.7. .y=1x sin x, y=1x2 sin x, y= xsin 1x, y= x2sin 1x, y=x sin 1x.(: 6, 7 83.10. 3.)8. 4.9, ; (xn). 6, 7 8 3.10 sin x= 1sin1x= 1 (0, +).limx+xsin x, limx+x2sin x, limx0+sin 1x , limx0+1x sin 1x . : a 0, limx0+xasin1x . 4.1474.9 . y= f(x) (a, ). , x (a, ) , . f(x), limxf(x) = +. f(x), u f(x) u x (a, ). f(x) , limxf(x)=, u. - limxf(x) = limxf(x) = +. : (i)(a, )(a, +), y=f(x) + (ii) (, b) (, b), y=f(x) . 4.11: . 4.1 . 4.1. (1) y= f(x) . (i) , y= f(x) , (ii) +, y=f(x) ., (i) , y=f(x) , (ii) , y=f(x) .(2) y=f(x) . (i), y=f(x) , (ii) , y= f(x) . , (i), y=f(x) , (ii) +, y=f(x) .4.1 ; 2.1 . 148. , 4.1 ..(1) 4.1 . ,a > 0, limx+xa= + limx0+xa= 0 y= xa (0, +). + . , limx+xa=. limx+(2x)a=limy+ya=. = limx+(2x)a= limx+ 2axa=2alimx+xa=2a, , =2a. =0 x 1 xa1a=1 , ,= limx+xa limx+ 1 = 1. +. 0. , xa>0 x>0, limx0+xalimx0+ 0=0, , -: limx0+xa= 0. limx0+(2x)a=limy0+ya=. =limx0+(2x)a=limx0+ 2axa=2a , ,= 2a. = 0.(2) y= (1 +1x)x (0, +). . ( 2 6.9). ,, , limx+(1 +1x)x +. (n), +, 4.9 _(1 +1n)n_ . , e, , e. ,limx+_1 +1x_x= e.,, limx+(1 +1x)x= e : , y= (1 +1x)x !! . (!) limx+(1 +1x)x= e. limn+(1+1n)n= e. , limn+(1+1n+1)n= limn+(1+1n+1)n+11+1n+1=e1= e limn+(1+ 1n)n+1= limn+(1+ 1n)n(1+ 1n) = e1 = e. >0, n0

e 0 [f(x) f()[ 0 > 0 x ( , + ) f(x) (x, f(x)) f() f() + , , ( , + ) y= f() y= f() + . 5.1: .. ,, .(1) y= p(x) = a0 +a1x+ +aNxN , limx p(x) = p().(2) y= r(x) =a0+a1x++aNxNb0+b1x++bMxM . , , , limx r(x) = r().153 5.2: .(3) y= cos x y= sin x , limx cos x = cos limx sin x = sin ., y= tan x y= cot x ; , ,=2+ k(k Z) ,=k(k Z), limx tan x = tan limx cot x = cot .(4) y= xa ; limx xa= a ,(i) , a ( =0, a 0) (ii) 0, a ( = 0,a 0).(5) a > 0,y= ax , limx ax= a.(6) a > 0, a ,= 1, y= logax . , > 0 limx logax =loga..1. 1. .y= x, y= 2x 3, y= x2, y=1x , y=x.2. 0;y=_sin xx, x ,= 01 , x = 0y=_1cos xx2, x ,= 012 , x = 03. ;y=_x, x 01x , x > 0y=_0 , x = 01|x| , x ,= 0y=_x2, x ,= 01 , x = 0y=_x2+ 1, x < 0x + 1 , x 0y=_x2, x x > sin x, < x 1544. ; 2 4.3.y= [x], y= [2x], y= x [x], y= x [x] 12 , y=x [x] 12.5. limh0_f( + h) f( h)_=0y=f(x) (a, b) a < < b. : y= f(x) =_1 , x = 00 , x ,= 0 = 0.6. y= f(x) , M [f(x)[ M x (a, ) (, b). (, > 0 > 0) y= g(x) = (x )f(x) .7. M 0 >0 y=f(x)(a, b),a 0) . y=f(x) , Hlder- Hlder- . =1, y =f(x) Lipschitz- . y =x, y = [x[, y =cos x, y =sin x, y =[x[ y =x[x[ Hlder- 0 Hlder-. Hlder- ,= 0 Hlder-. Hlder- = 0 ,= 0.8. y=f(x) (, f()) , , . ,, . .y=_x(1)[1x], x ,= 00, x = 0 . 1 3.4. 0.(: .) 0.5.2 .. 5.1 4.2.155 5.1. y= f(x) y= g(x) (a, b) a < < b [, b) (a, ]. ,, .. y= f(x) y= g(x) (a, b) a < < b, (a, )(, b) f()=g(). y=f(x) , limx f(x)=f(). y=f(x) y= g(x) (a, ) (, b), limx g(x) = limx f(x) = f(). , f() = g(), limx g(x) = g() ,,y= g(x) . [, b) (a, ] . 5.3: ..(1) y =_x + 1 , x 1x 1 , 1 < x y =x + 1 (, 1]. 1, , 1. 1 .(2) y=x2 y=_1 + x, [x[ 1010x2, [x[ < 1010 (1010, 1010). 0, 0.. 5.2 , , , . 5.2. y= f(x) y= g(x) , y= f(x) +g(x), y= f(x) g(x), y= f(x)g(x) y= [f(x)[ , . f(x)g(x) , g() ,= 0.. limx f(x) = f() limx g(x) = g() limx(f(x) + g(x)) =limxf(x) +limxg(x) = f() + g(),,y= f(x) +g(x) . , , ..(1) y =x+ex(x2x2) log x , y= x, y=ex, y= log x y=x 2x2 . (0, +) . (0,12) (12, 1) (1, +).(2) y=x2+xsin x+cos x 0 , 0 ,= 4+ k(k Z).156. . 5.3. z= g_f(x)_ y= f(x) z= g(y). y= f(x) z= g(y) = f(), z= g_f(x)_ .. >0. z=g(y) ,

>0[g(y) g()[ 0 [f(x) [= [f(x) f()[ 0 [g(y) g()[ 1x, [x[ 1 (, +) . . y= f(x) [a, b]. c, f(a) f(b), . c f(a) c f(b) f(b) c f(a) [a, b] f() = c, , c f(x) = c ( ) [a, b]. f(a)=f(b), c=f(a)=f(b), f(x)=c; =a =b. , f(a) ,=f(b) c=f(a)c=f(b), f(x) = c ; = a = b,. , f(a) < c < f(b) f(b) < c < f(a), c f(a) f(b), ., a b f(x) = c, (a, b). 5.9: . , , - . y= c c . (a, f(a)) (b, f(b)) f(a) f(b),,,,y= c . y= f(x), (a, f(a)) (b, f(b)), y= c. (, ), = f() , , = c y= c. f() = c. f(x) =c, . ,, .167 c. , . , , , ..(1) y= f(x) =_1 , 0 < x 10 , x = 0 [0, 1], 0 12 , c f(0) = 0 f(1) = 1, .(2)y=f(x)=_x, 0 x 0 x I f(x) < 0 x I. x-, x- x-... - .1. (0, 1);y= x2, y= x2x + 1, y= sin(x), y= cot(x), y= sin(2x).2. y= sin1x (0, +) . ; y= xsin x y=1x sin1x (0, +).(: 6,7 8 3.10.)3. y=11+x sin1x (0, +).(: y=11+xy= 11+x (0, +). 6,7 8 3.10.)4. () t = a t=b(al x[a, b]. > l f(x) x [a, b]. ;6. y= f(x) y= g(x) [a, b] f(x) > g(x) x[a, b]. f(x) g(x) + x [a, b]. ;(: y= f(x) g(x) [a, b].). .1. x73x6+ 5x5+ 13x4x312x25x + 1 = 0 [0, 1].1692. ex= x + 2 .3. 3x+2x1+1x2+5x3= 0 (0, 1), (1, 2) (2, 3).(: , .)4. tan x=x (2+k,2+ k)(k Z).5. y=f(x) [0, 1] 0 f(x) 1 x[0, 1]. [0, 1] f() = 2.(: y= f(x) x2.)6. y= f(x) y= g(x) [a, b]. f(a) < g(a) f(b) > g(b), (a, b) f() = g().(: y= f(x) g(x).) ;7. () . t = a t = b . . ;8. y=f(x) --I. y= f(x) I.(: , x1, x2 x3I x1 0, (, +) (0, +). :d axdx= axlog a.201 a>0, a ,=1, . y= ax x = logay,,d axdx=dydx=1dxdyy=ax=11log a1yy=ax= axlog a.a = 1, y= 1x= 1 . 1xlog 1 ,, . y= ex:d exdx= ex.. , y =xa a . y= xa [0, +), a > 0, (0, +), a < 0. d xadx= axa1_a > 1,x 0a < 1,x > 0._x > 0. y= xa= elog(xa)= ea log x . z= a log x, y= ezd xadx=dydx=dydzz=a log xdzdx= ezz=a log xax= ea log xax= xaax= axa1. a < 0, y= xa 0. a 0 < a < 1,d xadxx=0= limx0+xa0ax0= limx0+xa1= +. , a a > 1,d xadxx=0= limx0+xa0ax0= limx0+xa1= 0 = a0a1..1. .y= xlog x, y= loglog [x[, y= log_e3x2+4+ sin_x54__,y= 2x2+1log3(sin x), y= 3sin(log x), y= sin_elog(x2+1)_.2. .limx1log xx 1 , limx0ex1x, limx1xa1x 1. limx1log xxa1=1a , limx1xaxbx 1= a b, limx1xaxblog x= a b,limx0axbxx= log ab, limx0eaxebxx= a b, limx0tanh(ax)x= a.2023. y= f(x) y= g(x) (a, b), a < < bf(x) > 0 x (a, b). , y= f(x)g(x) . .y= xx, y= (x2+ 1)sin x, y= [x 1[x2[x 2[x1.4. d cosh xdx= sinh x,d sinh xdx= cosh x,d tanh xdx=1(cosh x)2 ,d coth xdx= 1(sinh x)2(x ,= 0).5. d arccoshydy=___1y21, y> 1+, y= 1d arcsinhydy=1y2+ 1,d arctanhydy=11 y2([y[ < 1),d arccothydy=11 y2([y[ > 1).6.7 . x = x(t) y= y(t) I t. C (x, y)=(x(t), y(t)) , t I, . I C . t ,(x, y) = (x(u), y(u)), (x, y) = (x(s), y(s)) . x=x(t) y=y(t) C, -, t. , ..(1) x = x(t) = t+ y= y(t) = t+ t (, +), , ,= 0. C. , , ,,, ., t, C- x y = . , (x, y) xy= , , ,= 0, t =1x t x = t + y= t +. , C (x, y) xy= . ,=0. , , x y= l : ,= 0, 203 y= y(x) =x +, ,= 0, x=x(y)=y +. , ,=0 ,=0, , . = 0, , x = , , . = 0,, y=, , .,, C l., l ax + by= c, a, b ,=0. = b, =a , =c. , C x = x(t) = t + y= y(t) = t + l.t = 0 t = 1 A0= (, ) A1= (+, +) A0A1= (, ) t. x = x(t) = t + y= y(t) = t + , (x, y)=(t + , t + )=(, )t + (, ). (, )(, ), , (, ) (t = 0) (, ).t (, +), (x, y)=(t + , t + ) .(2) x=x(t) =r0 cos t + x0 y =y(t) =r0 sin t + y0 t I= [t0, t0+2] 2, r0> 0, (x0, y0) r0 . , (x, y) = (r0 cos t + x0, r0 sin t + y0) (x x0)2+ (y y0)2= r02, , (x, y) (x x0)2+ (y y0)2=r02(x, y)=(r0 cos t + x0, r0 sin t + y0)tI. , , x = x(t) = r0 cos t +x0 y= y(t) = r0 sin t +y0 (xx0)2+(y y0)2= r02. , t I , (x, y) = (r0 cos t +x0, r0 sin t +y0) . t 0, (x0, y0) 20 20 . (x, y) = (0 cos t + x0, 0 sin t + y0) _x x00_2+_y y00_2= 1 . , x=x(t)=0 cos t + x0 y=0 sin t+y0 (xx00)2+(yy00)2= 1. t 204 I , (x, y) = (0 cos t+x0, 0 sin t+y0) . t < 2, (x, y) = (0 cos t+x0, 0 sin t+y0) .(4) y=f(x)I, (x, y)=(x, f(x))x I, . , x = t y= f(t) t I., x = g(y) I , x = g(t) y= t t I. . x=x(t), y=y(t) z=z(t) I t, (x, y, z) =(x(t), y(t), z(t)) , t I, ..(1) x = x(t) = t+, y= y(t) = t+ z= z(t) = t+ t (, +), , , ,= 0, . ,(x, y, z) = (, , )t + (, , ), (, , ) (t =0) (, , ).(2) x=x(t) =r0 cos t + x0, y=y(t) =r0 sin t + y0 z =z(t) =h02t + z0 t (, +), r0>0 h0>0, , . t 2, (x, y)=(r0 cos t + x0, r0 sin t + y0) xy-(x0, y0) r0, , z h0 . , (x, y, z) ,, r0 l xy- (x0, y0, 0). , , ., , x=x(t) y=y(t)I t. , , (x(), y()) I. +h I, h , , (x( +h), y( +h)) . x = x(t) =x(+ h) x()h(t ) + x(),y= y(t) =y(+ h) y()h(t ) + y(). h 0, (x(), y()), , , 205 :x = x(t) = x

()(t ) + x(),y= y(t) = y

()(t ) + y(). t : x

()_y y()_ = y

()_x x()_. (x

(), y

()) (x(t), y(t)) t = , , . x =x(t) =x

()(t )+x() y =y(t) =y

()(t )+y() x

(), y

() ,=0. , x = x(), y= y(). I, t , I, , . , x = x(t), y= y(t) z= z(t) I t, (x(), y(), z()) :x = x(t) = x

()(t ) + x(),y= y(t) = y

()(t ) + y(),z= z(t) = z

()(t ) + z(). (x

(), y

(), z

()) (x(t), y(t), z(t))t = . , x

(), y

(), z

() ,= 0..(1) x=x(t) =t cos t, y=y(t) =sin t t (, +) . (x(0), y(0)) = (0, 0) x = x

(0)(t 0) +x(0) = t, y= y

(0)(t 0) + y(0)=t. (t)y=x. (x

(0), y

(0)) = (1, 1).(2) x=x(t) =t, y=y(t) =t2, z=z(t) =t3t(, +) . (x(1), y(1), z(1)) =(1, 1, 1) x=x

(1)(t 1) + x(1) =t, y =y

(1)(t 1) + y(1) =2t 1,z =z

(1)(t 1) + z(1) =3t 2. (x

(1), y

(1), z

(1)) = (1, 2, 3).(3) y= f(x) (x I) x = t y= f(t) t I. (, f()) y= f

()(x ) + f() x (, +). x=2061 (t ) + y =f

()(t ) +f() t (, +). x=t y=f

()(t ) + f()t (, +), t, y= f

()(x ) + f() x (, +)., ..1. : .2. .3. () x=x(t)=r0 cos t + x0,y= y(t) = r0 sin t +y0 z= z(t) =h02t +z0 t (, +), r0> 0 h0> 0.4. () x=x(t) =r0t cos t + x0,y= y(t) = r0t sin t +y0 z= z(t) =h02t +z0 t (, +), r0> 0 h0> 0.5. a > 0. (x, y) x2y2= a2. - x =t2+ a2 y= t t (, +) x = t2+ a2 y= t t (, +). x=x(t) =a cosh t y =y(t) =a sinh t t (, +) x=x(t)= a cosh t y=y(t)=a sinh t t (, +). , . : , .6. (x, y) xy= b, b ,=0, x=x(t)=t y=y(t)=bt x=x(t)=bt y=tt(0, +) t (, 0). .7. x2 y2=a2(a>0) xy= b (b ,= 0) , .2078. x=x(t)=t3 y=y(t)=t4t (, +),, C (t3, t4). C (0, 0). , x = x(s) = s y= y(s) =3s4s (, +). C (0, 0);6.8 . y=f(x) (a, b) (a, b). y=f(x) (c, d), , , , f(x) f() x(c, d). y= f(x) [, b)(a, ), y=f(x) [, d) f(x) f() x [, d). , y= f(x) (a, ] (, b), (c, ] f(x) f() x (c, ]. y= f(x) f(x) f() f(x) f()., y=f(x) , , . y=f(x) . , , - () , . , , (), . y=f(x) : ( ) (, f()) ,, (, f())..(1) 0 y= 1 + x2(x + 1), 0 1 1+x2(x+1) 1 x (1, +) 0. 0 () , < 1. , 2 3 < 1 , , limx(1 +x2(x +1)) = .(2) 0 y= x x [0, +), 0 0 x x 0 [0, 1). 0 () , > 0.(3) , , . ,, y=_xsin1x, x > 00, x = 0208 [0, +). . , 6, 7 8 3.10, 7 8 4.8, 2 35.5, 6.2, 6 76.4 3,4 5 6.5. 0, limx0+xsin1x=0. x=12+n2(n Z, n 0) x sin1x=x>0x=132+n2(n Z, n 0)x sin1x= x < 0. n , 0. , [0, d), 0. 0 . .Fermat. y=f(x) (a, b) (a, b). y= f(x),(i) y= f(x) ,(ii) y= f(x) f

() = 0. 6.6: Fermat.. y= f(x). (c, d) (a, b) f(x) f() x (c, d). (i), f

(). f

+() f

() f

() . x(, d)f(x)f()x0. f

()=f

+()=limx+f(x)f()x0. , x(c, )f(x)f()x0. f

()=f

()= limxf(x)f()x0. f

()=0,, (ii). , , 0 0 . Fermat . y=f(x) y=f(x), (, f()) 0, .209.(1)0 () y= [x[, (, +), 0.(2) 0 () y =x2, (, +), 0 , , d x2dxx=0= 2xx=0= 0. Fermat . , : () , 0. .. y=2x3 9x2+ 12x + 5[0, 4]. y=d(2x39x2+12x+5)dx=6x2 18x + 12=6(x 1)(x 2), [0, 4] 0 4 1 2 . 5, 37, 10 9,. . [0, 4], . ,,0 ( 5) 4 ( 37). 1 2 . [0, 2] , . 0, 1 2 . 5, 10 9, 1 [0, 2] , , [0, 4]. , 1, 2 4 10, 9 37, ,2 [1, 4] , , [0, 4]. .Fermat . 6.4. (1) y= f(x) [, b) y= f(x). (i) f

+(),(ii) f

+() f

+() 0 f

+() 0,.(2) y= f(x) (a, ] y= f(x). (i) f

(),(ii) f

() f

() 0 f

() 0,.. (1) y= f(x). [, d) f(x) f() x [, d). (i), f

+(). f(x)f()x 0 x (, d) ,, f

+() = limx+f(x)f()x 0. (1) (2) . , f

+() 0f

+() 0, - f

+() = f

+() = +,. f

().210.(1) y=x[0, 2] 0 0, , 1 0. 2 2 1 0.(2)y= x[0, 2] 0 0 ,, + 0. Rolle. y= f(x) [a, b] (a, b). f(a) = f(b), (a, b) f

() = 0. 6.7: Rolle.. .(i) y= f(x) [a, b], f(a) = f(b), 0 (a, b), . y= f(x) [a, b], (ii) > f(a) = f(b) (iii) < f(a) = f(b). .(ii) - [a, b], y=f(x). f(a)=f(b), f() > f(a) = f(b) , , (a, b). ,y= f(x) , Fermat, f

() = 0.(iii) - [a, b], y=f(x). 0 limx+f

(x) = 0. x [x, x + 1] [f(x + 1) f(x)[ < .)18. y= f(x) [, b) (, b). limx+f

(x), f

+() .(: >0 limx+f

(x)=. x [, x] f(x)f()x < .) y= f(x) (a, ] (a, ). limxf

(x), f

() .6.9 .6.9.1 . 6.5. y= f(x) I.(1) y= f(x) I f

(x) = 0 x I.(2) y= f(x) I f

(x) 0 x I.(3) y= f(x) I f

(x) 0 x I.. (1)y=f(x) , . , f

(x)=0 xI. x1 x2Ix10 f

(x) = +. , f

(x) 0 f

(x) < 0 f

(x) = . 6.7 .(1) (2)6.6. y=f(x) , , f

(x) 0 x I. y= f(x) .. y=x3 (, +) f

(x)>0 x (, +). :d x3dx= 3x2 x > 0 x ,= 0 0 x = 0. 6.5 6.6 . , ..(1) y=|x|x , (, 0) (0, +), (, 0) (0, +). 1 (, 0) 1 (0, +).(2)y=1x 1x20. (0, x) exx1x=f(x)f(0)x0= f

() = e1. >0, e 1 >0, , ex x 1 >0. , , x < 0, (x, 0) exx1x=f(x)f(0)x0= f

() = e1. < 0, e1 < 0 ,, exx 1 > 0., , x ,= 0 ex> x + 1 x = 0,, e0= 0 + 1.. . y=f(x)=ex x 1 f

(x)=ex 1> 0 x > 0 < 0 x < 0. [0, +) (, 0]. f(x) > f(0) = 0 x ,= 0.218(3) cos x 1 x22 x. . y= f(x) = cos x 1 +x22 (, +). f(0)=0 x ,=0. (0, x)(x, 0)cos x1+x22x=f(x)f(0)x0= f

() = sin . x > 0, > 0, , > sin . , x 0 (0, +) < 0 (, 0). [0, +) (, 0] , , f(x) > f(0) = 0 x ,= 0... .1. .y= x2x 1, y= x315x2+ 72x + 7, y=x22x + 3x2+ 2x + 3 , y=xx + 4 ,y= x2ex, y= sin x cos x, y= sin(3x)3cos x, y= x + sin x,y= x +[ sin x[, y= log xx, y= [x[e|x1|, y= arctan x log(1 + x2) .2. y=_1 +1x_x (0, +).3. y=_x x2sin1x, x ,= 00, x = 0 d ydxx=0= 1 > 0. a>0, , (a, a).4. -.(i)y= (x 1)[x[ [1, 3].(ii)y= [x23x + 2[ [3, 10].(iii)y=(log x)2x[1, 3].(iv)y= x +1x[13, 3].(v)y= exsin x [0, 2].5. a > 0 y= a log x+2ax (0, +) .6. a1 f() f(x) < f() .(: f

() 4.6.) 3. y= f(x); f

() < 0;10. y =f(x) I f

(x) ,=0xI. 11y= f(x) -- I.(1) y= f(x) I.(: : 8 5.5. : Fermat , x1, x3 Ix1< x3 y=f(x)[x1, x3] x1, x3 . x1, x2, x3 I x1 0 > f

(b), 13 14 .)11. . l M=(x0, y0) . M l M l d(M, l).(i) l x = , d(M, l) = [ x0[.(ii) l y= x + , d(M, l) =