EBOOK bài tập đại số và GIẢI TÍCH 11 PHẦN 2 vũ TUẤN (CHỦ BIÊN)

%ieangIV GI6IHAN §1 Gidi hqn cua day so A KIEN THOC CAN NHd Gidi han hum han • lim M„ = va chi IM„I cd thi nhd hon mdt sd' duong be y, kl tfl mdt sd hang nao dd trd di • lim v„ = a lim (v„ - a) = n—>+oo n->+oo Gidi han vo circ* • lim M„ = +00 va chi M„ cd thi ldn hon mdt sd duong ldn y, n—>+oo kl tfl mdt sd hang ndo dd trd di • lim M„ = -00 •« lim (-M„) = +c» n—>+oo ^ n—>+ n vdi mgi n Chflng minh ring limM„ = +00 Gidi Vi lim/2^ = +00 (gidi han dac biet), nen n cd thi ldn hon mdt sd duong ldn y, kl tfl mdt sd hang nao dd trd di 142 Mat khdc, theo gia thidt M„ > n vdi mgi n, nen u„ cung cd thi ldn hon mdt sd duong ldn y, k l tfl mdt sd hang ndo dd trd di Vdy lim M„ = +OO ^ Nhan xet : Trong cac vf du tren, ta da van dung true tiep cac djnh nghTa ve gidi han cCia day sd • Vidu An^ Tinh lim -n-l + 2/2^ Gidi An"- Ta cd lim -n-l 4-i-L = lim- + 2/2^ n 4- n = n • ViduS Tfnh lim yj3n^ + + n 1- -2/2^ Gidi , — , Tl n\3 + —r^ + n / 1 3+— +— n n' n\ l-2rf 1-2/2^ -2 • Vidu Tfnh lim n - h+l Gidi lim n n+l = lim n^ + n^ -2 = lim n+l n n ^ = +00 n 143 • Vidu7 -i_ Tfnh lim(-/2^ + n4n + 1) Gidi lim(-/2^ + n4n + 1) = lim{-n^) • I- A V/2 ; n' — —00 ViduS -p Tfnh lim •\//2 + n - -)• Gidi , ^ ^ (V/2^ +/2 - V/2^ - (v/2^ +/2 + V/2^ - lim V/2^ + /2 - V/2^ - = lim-^^ '^ '^ ' yjn^ +n+ yfn^ - n+l = lim y/n^+n + yln^ ^ l+• = lim- n • = lim I 1.1 LX ' nJl+— + nJl—- Luu y : Khi giai bai toan Vf du 7, ta da bien ddi ve dang cd thd ap dung hai tinh chat sau : • limM„ = + 00 lim(-M„) = -OO (1) • Neu limM„ = +oo v^ limv„ = a > thi limM„v„ = +oo (2) Tuy nhien, nhOng bien ddi tren Ichdng cdn thfch hgp vdi Vf du Qu^ thirc, ndu lam tuong tu nhu vay ta se cd : limNn^ +n-^rf - 1 = lim nAl + nAl — - V = lim/2 r V Vi lim 144 -HH 1+- 1n = 0, nen khdng the ap dung tinh chat (2) d tren ^ Nhan xet: De tim gidi han cCia mot day sd ta thudng dUa ve cac gidi han dang dSc biet va ap dung cac dinh li ve gidi han hOu han hoac cac djnh li ve gidi han vd cue De cd the ap dung dUdc cac djnh If ndi tren, thong thudng ta phai thUc hien mdt vai bien ddi bieu thflc xae dinh day sd da cho Sau day la vai ggi >^ bien ddi, cd the van dung theo tflng trUdng hop : - Ndu bieu thflc cd dang phan thflc ma mau va tfl deu chfla cac luy thfla cOa n, thi chia tfl va mau cho n , vdi k la sd mu cao nhat - Neu bieu thflc da cho co chfla n dudi dau can, thi cd the nhan tfl sd va miu sd vdi cijng mot bieu thflc lien hgp • Vidu Cho day sd (M„) xae dinh bdi • Ml = V "n+l = v ••" "n Vdi/2>1 Bilt (M„) cd gidi han huu han 1dii n —> +00, hay tim gidi han dd Gidi Ddt limM„ = a Ta cd ''n+l ^2 + M„ => limM„+i = lun y/2 + M„ => a = yf2 + a =>a -a-2 = 0^^a Vi M„ > nen limM„ = a > VdylimM^ = = -l hoac a = Luu y : Trong Idi giai tren, ta da dp dung tfnh ehd't sau ddy "Nlu limM„ = a thi limM„+i = a" Ban dgc cd thi chiing minh tfnh chdt bing dinh nghia • Vidu 10 Cho day sd (M„) xdc dinh bdi cdng thflc truy hdi "1 = M„^i *n+l = 2-M„ vdi n >l Day sd (M„) cd gidi han hay khdng n^> +(p'? Ndu cd, hay tim gidi han dd 10 BTBS>11-A 145 Gidi Ta cd Ml = — ; M2 = :r- ; M3 = — ; M4 = — Tfl dd dir dodn u„ = -.(1) Chflng minh dfl dodn trln bing quy nap : - Vdi /2 = 1, ta cd MI = - — - = - (dung) - Gia sfl dang thflc (1) dflng vdi /i =fe(fe > 1), nghia la MJ^ = Khi dd ta ed M^+I = = fe + r— = -r—^, nghia la dang thflc (1) ^"feTT cung dflng vdi n-k+l - Vdy u„ = — ^ V/2 e N* " n+l Tfl dd ta cd limM_ = lim = lim = " n+l , 1+ — n ^ Nhan xet : De tim gidi han ciia day sd cho bang cdng thflc truy hdi ta cd the tim cdng thflc tdng quat, cho phep tfnh u„ theo n, bang each dfl doan cdng thflc nay, va chflng minh du doan bang quy nap Sau dd, tim gidi han cua (i2„) qua cdng thflc tdng quat • Vidu 11 Giai Day sd vd han , - V2,1, —j=, —, la mdt cdp sd nhdn vdi cdng bdi -yf2 146 10 BTDS>11-B Vi \q\ = yfi = —j= < nen day sd la mdt cdp sd nhdn lui vd han V2 \ 2V2 + J _ " V + l' Dodd,5=2-V2 + l - = + yfi • Vidu 12 l i m dang khai triln cua cdp sd nhdn lui vd han (v„), bilt tdng cua nd bing 32 va V2 = Gidi Tfl gia thidt suy , ^ = 32 Mat Idiac, V2 = Vi Vi = — The vao dang thflc tren ta cd : —= 32 — >- —, —, V 2y 1.12 Tfnh tdng = + 0,9 + (0,9)^ + (0,9)^ + + (0,9)"n-l' + 1.13 l i m sd hang tdng qudt cua cdp sd nhdn lui vd han cd tdng bdng va cdng bdi^=f 1.14 Cho day sd' (6„) cd sd hang tdng quat la b„ = sina + sin a + + sin"a n vdi a^ — + kn l i m gidi han cua {b„) 149 Hinh Do thi hdm so y = sin4x + b) Vi sinAx + l = m •» sin4x = m-l va —l limx„ = l ( - ) = - 21 a) Day (x„) bi chan vi n 5/22 < -^; < voi moi n ; rf +3 b) Day (y„) bi chan vi 2/2 \yn\ = (-1)" n+l e) Day (z„) khdng bi chdn vi |z„| = l/2COSrtTCl = /2 234 , , I sia/21 < 2/2 7 ^„ Mat khdc, day sd bi chdn tren vi x„ •" 1 1 1 -T + - + = - r + + + 5" + -oo [ • x|4 + - 25 a) lim smx - sma x-ya X-a x-a X +a cos —-— sm —-— 2 Um = cosa x-a b) Ddt - X = r (r ^ X -)• 1), ta cd lim(l - x)tan-— = lun rtan(l - i)— x-^\ x->0 1 = Um/cot—f = Um ——^ 2->0 f->0 ^ 71 tan-r 236 2^ Tl" = +00 ^ Chu y lim tanx Jc->0 X lim sinx X COSX Jf->P = c) + V3 5-3V3' d) , t a n x - s i n x , 1-cosx Um = lim — I x-*o sin X x^o sin x cos x a) 2(1 - 2x) 2sin^= lim x^O • 2X 2X 4sm —cos —cosx 2 1 ( l - X + x2)2 b) 12 - 6x - 6x2 + 2x^ + 5x^ - 3x^ (1-^)^ c) d) e) f) -2eosx(l + sinx) + cos X 2sin^x 2x X -—sin-^i-tan— 3 cosI 2x - — n sm x - 27 A = Khi dd/(x) ed dao hdm tai x = 237 MUC LUC Trang Chumg / HAM SO LUDNG GIAC - PHUGNG TRINH LlTONG GIAC § Hdm sd Iflgng gidc §2 Phuong trinh lugng gidc CO ban 13 §3 Mdt sd phuong trinh lugng gidc thudng gap 24 Bdi tdp dn chuong I 35 Ldi giai - Hudng ddn - Dap sd chuong I 36 Chuang II TO HOP - XAC SUAT 57 §1 Quy tic ddm 57 §2 Hodn vi, chinh hgp, td hgp 60 §3 Nhi thflc Niu-ton 64 §4 Phep thfl va bidn cd 66 §5 Xdc sudt cua bidn cd 69 Bdi tdp dn chuong II 73 Ldi giai - Hudng ddn - Dap sd chuong II 74 Chuang IIL 238 DAY SO - C ^ SO CONG VA CAP SO NHAN 87 § Phuong phdp quy nap toan hgc 87 §2 Day sd 96 §3 Cdp sd cdng 107 §4 Cdp sd nhdn 114 Bai tdp dn chuong III 121 Ldi giai - Hudng ddn - Dap sd chuong III 124 Chumg IV Gldl HAN 140 § Gidi han cua day sd 140 §2 Gidi han cua ham sd 151 §3 Ham sd lidn tuc 160 Bdi tdp dn chuong IV 165 Ldi giai - Hudng ddn - Ddp sd chuong IV 170 Chumg V DAO HAM 190 § Dinh nghia vd y nghia eua dao ham 190 §2 Cdc quy tic tinh dao ham 195 §3 Dao ham cua cae ham sd lugng giac 199 §4 Vi phdn 203 §5 Dao hdm cdp hai 205 Bdi tdp dn chuong V 207 Ldi giai - Hudng ddn - Dap sd chuong V 209 On tdp cud'i ndm 220 239 Chiu trach nhiem xudt bdn : Chu tich HDQT kiem T6ng Giam d6c NGO TRAN AI Pho Tdng Giam d6'c kiem T6ng bidn tap NGUYfeN Q U t THAO Bien tap ldn ddu: NGUYfiN XUAN BINH - NGUYfiN NGOC TU Bien tap tdi bdn : NGUYfeN XUAN BINH Bien tap Id thuat: NGUYfeN THANH THUt - TRAN THANH HANG Trinh bdy bia : TRAN THUt HANH Sih bdn in : Lfe THI THANH HANG Che bdn : C N G TY CF THifeT KE VA PHAT HANH SACH GIAO DUC BAI TAP DAI SO VA GIAI TICH 11 Ma so : CB103T1 In 35.000 cudn, khd 17 x 24 cm In tai Cdng ty TNHH MTV in Quang Ninh Sd in: 2134 So xuat ban: 01-2011/CXB/824-1235/GD In xong va nop luu chieu thang nam 2011 240 •u< V I / O N G M I E N KIM CUdNG CHAT LUONG QUOC TE HUAN CHKONG HO CHI MINH SACH BAIlAPLCfP 11 BAI TAP DAI so VA GIAI TICH 11 BAI TAP HiNH HOC 11 BAI TAP NGO" VAN 11 (tap mot, tap hai) BAITAPVATLIU BAlTAPUCHSCfll BAI TAP HOA HOC 11 10 BAI TAP TIENG ANH 11 BAI TAP SINH HOC 11 11.BAITAPTIENGPHAP11 B A I T A P O I A U ' H 12 BAI TAP TIENG NGA 11 BAI TAP TIN HOC 11 SACH BAI TAP LCfP 11 - N A N G CAO BAI TAP OAI so VA GIAI TiCH 11 , BAI TAP HOA HOC 11 BAI TAP HiNH HOC 11 , BAI TAP N G U V A N 11 (tap mot, tap hai) BAI TAP VAT LI 11 , BAI TAP TIENG ANH 11 Ban doc co the mua sach tai: • • • • Cac Cong ty Sach - Thiet bi truong hoc a cac dia phucmg Cong ty CP Dau tu va Phat triSn Giao due Ha Noi, 187B Giang Vo, TP Ha Noi • Cong ty CP Dau tu va Phat trien Giao due Phuang Nam, 231 Nguyen Van Cu, Quan 5, TP HCM Cong ty CP Dau tu va Phat trien Giao due Da Nang, 15 Nguyen Chi Thanh, TP Da Nang hoac cac cua hang sach cua Nha xuat ban Giao due Viet Nam : - Tai TP Ha Noi : - Tai - Tai - Tai Tai 187 Giang V6 ; 232 Tay Son ; 23 Trang Tien ; 25 Han Thuyen : 32E Kim Ma ; 14 Nguyen Khanh Toan ; 67B Cua Bae TP Da Nang : 78 Pasteur ; 247 Hai Phong TP H6 Chi Minh 104 Mai Thi Luu ; 2A Dinh Tien Hoang, Quan ; 240 Tran Binh Trong ; 231 Nguyin Van Cir, Quan TP can Tho : 5 Duong 30'4 Website ban sach true tuyen : www.sach24.vn Website: www.nxbgd.vn 934994 " 023658 Gia: 12.400d [...]... {n -^ +oo) : , , (-3)" + 2. 5" a) lim^^— ; 1-5" ' 'ux ,• 1 + 2 + 3 + + /I b) lun ; rf +n + \ e) liml V /2^ + 2/ 2 + 1 - ^rf +n-\\ 2 Tim gidi han cua day sd (M„) vdi ^ (-1)" u^ 2" - /2 ^^ "« = ~2~ ~T ' b) M„ = — — - /2^ + 1 3" + 1 3 Vilt sd thdp phdn vd han tudn hodn 2, 1 3113 1131 (chu ki 131) dudi dang phdn sd 165 Ml = 1 4 Cho day sd (M„) xae dinh bdi 2M„ + 3 u„^, "n+l = w„ + 2 vdi /2 > 1 a) Chflng minh ring... + 5 tai X = 4 ; '_x-l I — ndu X < 1 b)^(x) = V2-X-1 taix=l -2x , ndu X > 1 3.6 Xlt tfnh lien tue cua cdc ham sd sau trdn tdp xae dinh eua chflng 1-x x^ -2 f^, ndu x^ y ]2 a)/(x) = X - V2 2^ , ndu x=y /2; b) g{x) = {x-2f 3 -, nduxTi 2 , ndu X = 2 x^ - X - 2 3.7 lim gia tri cua tham sd m di hdm sd f{x) = > x - 2 m , ndu X 9t 2 , ndu X = 2 lien tuc tai x = 2 Vx-1 3.8 Tim gia tri cua tham sd m di hdm sd... x ^2 Vx + 7 - 3 ^, , Vx^ - X - ylAx^ + 1 d) lim r——5 2x^ + 3x - 4 x^ + 1 J:->+«) —X AT—>-00 e) Um - | - ^ - l x-^o'-^V-^ + l J f) ZX + J Um {\1AX^ - X + 2x) Gidi , , x ' ^ + 2 x - 3 , ( x - l ) ( x + 3) , x + 3 4 a) U m — = lun — = Um =- -12x2-x-1 -i2(;,_i)(^ + | ) >i2x + l 3 2- X (2 - x)(Vx + 7 + 3) ^ J-^ = lim ^-^; ^ = l i m - (Vx + 7 + 3] = X - 2 ;c- >2 J : ^ 2 VX + 7 - 3 A: ^2 b) lim -^^ J , , 2. .. (2) Mat khae, lim 5" = +oo (3) 5" Tfl (1), (2) va (3) suy ra lim(5" - cosyfnn) = lim5" yfnv COSV /27 1 = +00 5" 171 Ml = 2 1.10 M„^, *«+l = -'^ Tacd, Ml = 2, Du dodn, M„ = v o i /2 > 3 "2= 2'' 5 "3 " 4 ' 9 "4=g- 17 "s = j ^ - 2" -^ + 1 ;— vdi n e N* /^n-l Chflng minh du dodn trdn bing quy nap (ban dgc tu chiing minh) r IY"! Y 2" "^ +1 Tfldd, IimM_ = lim ;— = lim 1 + = lim 1 + 2 1 " 2" "^ = 1 1 .11 f 1. 12. .. n—>+oo (2) n-^+ n V i , - - < 0 ndn f{b„) = Dodd, lim f{b^)= lim -L-i f4-i ) = - 1 (3) n-^+\n Tfl (1), (2) va (3) suy ra/(x) khdng cd gidi han khi x —> 0 n + 2nn {n e 2. 3 a) Xlt hai day so («„) vdi a„ = 2nn va {b„) vdi b^ = — Ta cd, lima„ = lim2nn = +oo ; limZ/„ = lim n + 2nn = lim /2 | ^ + 2 2 i | = +; \' lim sin a„ = limsin2 /27 i = limO = 0 ; Umsin6„ = lim sin — + 2nn = liml = 1 V2 y Nhu... 2n+l b) Dudodn : S„ -> :r-5^gc khi « ^ +°°, bay lim5„ = — C /22 ing/n2 /2/ 2.-5„ = Mi + M2 + + M „ = ^ + ^ + + ^ = i - - i ^ Tfl dd, lim5„ = - 2 2.1 a) - 4 ; 2. 2 /(x) = b) +oo x^ , nlu X > 0 x^ - 1, nlu X < 0 a) (H.6) Du dodn : Hdm sd/(x) khdng ed gidi han khi x -> 0 b) Ld'y hai day sd ed sd hang, tdng qudt la a„ = — vab„= Ta ed, a„-> 0 vd &„-> 0 khi /2- >+00 (1) ^- 173 V i - > 0 nen / ( a „ ) = ^2. .. ed nghidm ; b).cos2x = 2sinx - 2 cd ft nhd't hai nghidm trong khoang ;) Vx + 6x + 1 - 2 = 0 cd nghiem ducmg 164 6'"'' 3.10 Phuong trinh x"* - 3x^ + 1 = 0 ed nghidm hay khdng trong khoang (-1 ; 3) ? 3 .11 Chiing minh cdc phuong trinh sau ludn ed nghidm vdi mgi gia tri eua tham s6 m : a){l-m^){x+lf+x^-x-3 = 0; b) / 72( 2eosx - V2 ) = 2sin5x + 1 3. 12 Chflng minh phflong trinh x" + flix" + a2x" + + a„-ix +... x-^+00 _;c - X +1 x^-^ yfx^-yl^Ax^ d) lim 2x+3 +1 \x\Jl IxL 4 + —r = lim - 2x+3 jr->-oo - x J l - - + x j 4 + —r = lim - 4_ "^r2 v3 ^ ^ = -2 i 1 1 ^ x^ 2x+3 -Jl-i 4.J = lim - 2+ 1 X r e) lim — -1 ;c->0" X x + 1 156 l - ( x + l) -1 = lim = lim = -1 ;t^o- x{x + 1) ;,^o- {x + 1) 2 .2 „^ A Jl f) lim ( V 4 x 2 - x + 2 x ) = lim ^^^ ""^ "^"^ ^^^ ^^-^ yJAx^ - X - 2x ^ = lim , X-^^x> j = lim 1 , ;c->-oo /... a nn _ nn 17 lim bing 2" + 1 (A)l; (B)-a); (C) 0 ; (D) + 00 18 lim V /2^ -n + l - n\ bing (A) 0 ; (B) 1 ; (C) - ^ ; 2 ' (D) - 00 19 lun (x - x^ + 1) bing (A) ; 1 (B) - o ) ; (C) 0 ; (D) + 00 x-1 20 lim x ^2~ bing X -2 (A)-n+l - Vdi /2 = 1, mdt hinh vudng dflge tao thdnh ed didn tfch Id MJ = —- 1 72 (1) Vdy (1) dung - Gia sfl cdng thflc (1) dung vdi n =fe(fe > 1), nghia la M^ = —;—;-• Ta cdn 2 chflng minh (1) dflng vdi /2 =fe+1,tfle la chflng minh M^+I = 2^ +2 Thdt