Toán cao cấp cho các nhà kinh tế - Phần 2 - Giải tích toán học - Lê Đình Thúy

TRlTdNG DAI HOC KINH Ttz QU6c DAN

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-LE DINH THUY

TOAN CAO CAP

CHO CAC NHA KINH TE

PHAN 2: GIAI TICH TOAN HOC■

(Tai ban Ian thft ba, co chinh sua bo sung)

DAI HQC NGAN HANGTP HO CHI MINH

THU VIEN

NHA XUAt BAN DAI HQC KINH t£ Qlldc DANNAM 2012

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Cudn sdch nay Id cudn sdch thit hai ciia bo sdch hai tap

‘TOAN CAO CAP CHO CAC NHA KINH TE’, ditqc bien soan

dtfa theo chitcmg trinh mon Toan Cao cap cua fritting Dai hocKinh te qudc dan, dung chung cho cd hai khdi: Kinh te hoc vdQuan tri kinh doanh.

Tiep theo cudn Toan Cao cap cho cdc nhd kinh te, Phan 1:Dai so tuyen tinh’ da ditqc xudt ban, cudn Toan Cao cap chocdc nhd kinh te, Phan 2: Giai tich toan hoc’ bao ham nhitng noidung sau day:

Chitcmg 1 : Ham so vd gitii han.Chitcmg 2 : Dao ham vd vi phdn.Chitcmg 3 : Ham sdnhieu bien so.

Chitcmg 4 : Cite tri cua ham nhieu bien.Chitcmg 5 : Tich phdn.

Chitcmg 6 : Phitcmg trinh vi phdn.Chitcmg 7 : Phitcmg trinh sai phdn.

Trong pham vi cua cdc chitcmg noi tren, chiing toi Ian hteft trinhbay cdc van de ccf ban cua Giai tich Toan hoc, ciing vtii mot so noidung chon loc cua ly thuyet phitcmg trinh vi phdn vd phitcmg trinhsai phdn Nhitng noi dung do ditac hta chon can cit vao nhu can sit

TOAN CAO CXPCHOCACNHA KINH " '1

dung cdc phicang phdp Todn hoc trong phan tich kinh te, vai muc" dich trang bi kien thi'cc todn hoc can thiei de sinh vien co the tiep can

vai phicang phdp mo hinh trong Kinh tehoc thifc chieng.

Chuang 1 trinh bay nludig khdi niem caban ve ham sd'va giaihan, trong do cluing toi nhdn manh viec sic dung quan he ham so debien dien quan he giifa cdc bien so kinh te Ly thuyet giai han dicqctrinh bay mot each co he thong dot vai day so Doi vdi ham so ddi solien tuc cluing toi chichu trong dinh nghia bang ngon ngic day so, ticdo trinh bay tom tat (khong co chftng minh) cdc dinh ly ca ban vegiai han, khdi niem ham so lien tuc vd cdc tinh chat ca ban ciia hamlien tuc.

Cdc chicang 2,3,4 bao quat cdc noi dung ca ban ve phep todn viphan, pluic vu cho viec phan tich finh so sdnh trong kinh te Trongkinh tehoc, dqo ham vd vi phan dicqc sic dung dephan tich xu hicangthay ddi cua cdc quyei dinh khi dieu kien ngoai sinh thay ddi Cdcbai todn ciCc tri giic vai tro quan trong trong viec phan tich su hcachon toi itu ciia cdc chu the kinh te, do do, ciing vai bai todn cite tricua ham mot bien da dicqc trinh bay a chtcang 2, cluing toi ddnhrieng chicang 4 de trinh bay Ian heat cdc bai todn ciCc tri vd ieng dungtrong phan tich kinh te Ngoai ra, trong nhieu cud'n sach kinh te hocngicdi ta dung den dqo ham ciia ham an de phan tich cdc quan he,do do trong chicang 3 chiing toi co trinh bay mot each het sice cddong khdi niem ham an vd phicang phdp tinh dqo ham ciia ham an.

Cdc chicang 5, 6, 7 la ca sd todn hoc de tiep can vai viec phantich dong thai ciia cdc bien so'kinh tetheo thdi gian Cdc noi dungdicqc hca chon chi dieng a mice do sic dung ciia sinh vien bqc cic nhdnkinh te, khong chii trong ly thuyet Phep todn tich phan dicqc trinh

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-4*TrirdngDaihpcKinhte QUotdan

bay gian hcqc, chu yea la cdc phicang phap tinh loan Cdc chicang ve •phicang trinh vi phan va phicang trinh sal phan chi diCng a cdc khdi

niem ca ban nhat va phicang phap giai mot so loqi phicang trinh capmot, phicang trinh tuyen tinh cap hai va each phan tich dinh tinh quydqo thdi gian bang do thi.

Trong khuon khd cua cudn sdeh nay, cdc nqi dung toan hocdicac trinh bay bang ngon ngic danh rieng cho sinh vien cita cdctricdng thuqc khdi kinh te Dan xen vdi cdc nqi dung toan hoc,chiing toi diCa vdo mot so khdi niem ca ban cua Kinh te hoc, trenccf sd do Ian heat giai thieu cdc mo hinh sit dung toan hoc trongphan tich kinh te Gido trinh nay khong khai thde sdu cdc van deloan hoc ma chi de cap d mice do phuc vu cho mot cong cunghien cicu kinh te Cdc khdi niem toan hoc tricu ticang dicacdidn dat bang ngdn ngit mo td, giup ban doc ndm dicac thicc chatcua van de Da so cdc dinh ly dicac phat bieu khong co chicngminh ma chi hicang dan each sic dung Mac du vdy, tinh chat hethong va tinh chat che cua logic suy luan van dicac ton trong.Can noi them rang cdc nqi dung kinh teducfc diCa vdo Id nhdmmuc dich minh hoa, giup sinh vien lam quen vdi viec sic dungcong cu toan hoc trong phan tich kinh te Day la cudn sdeh toanhoc phuc vu cho cdc nhd kinh te chic khong phai la tai lieu phobien kien thi'Cc kinh te.

Vdi cudn gido trinh nay, chiing toi da cd gang tao dieng motcan true mon hoc de viec gidng day toan hoc co y ngliia thietthicc han ddi vdi viec ddo tao can bq trong linh vice khoa hockinh te va quan tri kinh doanh Chac han van con nhfeng van decan phai dicac tiep tuc thdo luan ve edit true cua cudn sdeh va

frirdng Daihoc Kinh te Qud'c dan5,

r TOAN CAO gApchocAcnhakinh

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cdc npi dung da dicqc de cap Chung toi mong moi co ditetenhftng y kien dong gop tic phia cdc nhd toan hoc, cdc nhd kinhte, cdc ban dong nghiep va dong dao ban doc de cd diCcfc motcuon gido trinh tot hern.

Cudi ciing, tdc gid xin ditete bay to long biei cm chan thanh tcriHoi dong Gidm dinh gido trinh ciia Triccmg Dai hoc kinh te qudcdan, cdc nhd gido ciia Bp mon Toan ccf ban va cdc ban dong nghieptrong va ngodi triccmg da co nhicng y kien dong gop het site bo ich,giup tdc gid hoan thanh viec bien scan cudn sdeh nay Xin tran trongcam cm NXB Dai hoc Kinh Te Quoc Dan da tao dieu kien dita cuonsachdenvaibandoc.

hocKinh

Chuang 1: Ham sova gi& han

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HAM SO VA GlOl HAN

a Khdi niem bien so

Trong cac linh vuc khoa hoc, chung ta thuomg gap cac dai lugngdo dirge bang so Khi nghidn cuu quy luat thay doi tri so cua cacdai lirgng do, ngirdri ta thuomg diing chu dd ky hidu sd do cuachung Chang han, trong hinh hoc, chung ta thirdmg dung chu Vdd ky hieu thd tfch Vdi mdi khdi hinh hoc trong khdng gian, Vla mdt sd thuc Ngucd ta goi V la mdt bien sd hinh hoc.

Trong ngdn ngtr hinh thuc cua toan hoc, tir “bidn sd” dugc hieutheo nghia nhu sau: -

Dinh nghia: Bien so U mdt ky hidu ma ta cd thd gan cho ndmdt sd bat ky thudc mdt tap sd X * 0 cho tnrdc (X cR) Taphgp X dugc goi la mien bien thien (MBT) va mdi sd thuc XqgXdugc goi la mdt gid tri cua bidn sd do.

dugc ky hidu bang cdc chu edi: x, y, z, Thdng thuomg ngubi tachi xet cac bidn sd md MBT cua nd cd ft nha't hai sd Mdt bidnsd chi nhan mdt gid tri duy nhat dugc goi Id hang so.

Trong giai tfch toan hoc ta thuomg xet cac bidn sd thay ddi gia trimdt each lidn tuc, vdi MBT Id mdt khoang sd Cdc khoang sddugc k^ hidu nhu sau:

T( )AN CAO CAP CHO CAC NHA KINH T&Khoang dong (doan): [a;b] = {x: a<x<b).Khoang mo: (a;b)={x: a<x<b).Cac khoang nua ma: [a; b) = {x: a < x < b |;(a; b] = {x: a<x <b).Cac khoang vd han: (-co; b] = {x: x<b};

(-oo;b)={x: x<b};[a; + oo) = {x: x>a};(a; + oo) = {x: x>a};(- 00 ; 4- oo) = R.

b Cac bien so kinh te

Trong Imh vuc kinh td ngucfi ta thudmg quan tam d6h cac dailuqng nhu: gia ca, luqng cung, luqng c£u, doanh thu, chi phi, thunhap quoc dan, ty 16 lam phat, ty 16 that nghi6p, Khi phan richxu hudng thay doi tri so cua cac dai luqng do theo khdng gian,then gian va theo cac dieu ki6n khac nhau, cac nha kinh td xemchung nhu cac bien sd Cac bien sd d6 duqc goi la bien sd'kinh te.Trong cac tai li6u kinh td, ngudri ta thubng ky hi6u cac bidn sdkinh td bang cac chu cai d£u cac th tieng Anh li6n quan ddn ynglua cua cac bieh sd do Sau day la m6t sd ky hi6u thucfng gap:

p : Gia hang hoa (price);

Qs : Luqng cung (Quantity Supplied);Qd : Luqng c&i (Quantity Demanded);U : Lqi ich (Utility);

TC : Tong chi phi (Total Cost);

TR : Tdng doanh thu (Total Revenue);

Y : Thu nhap qudc dan (National Income);C : Ti6u dung (Consumption);

S : Tiet ki6m (Saving);I : D£u tu (Investment).

8TrUdriij Dai hgc Kinh teQy&cjan J.1I !

: Ham so va gidi han < •

II.QUAN HE HAM SO

a Khdi niem ham so

Khai niem ham sd dirge sir dung rong rai trong nhidu linh vuc debidu dien quan he chi phoi Iln nhau giua cac bien sd Dinhnghia khai niem ham sd bang ngdn ngtr hmh thiic cua toan hocco ndi dung nhu sau:

Dinhnghia: Mdt ham sd'f xac dinh tren mdt tap hop Xc R la

mdt quy tac dat tirong ling mdi sd thuc xeX vdi mdt va chi mdtsd thuc y.

Tap hop X dugc goi la mien xac dinh (MXD) cua ham sd f Sd ytuong ung voi sd x, theo quy tac f, dugc goi la gid tri ciia ham so

Khi ndi deh cac ham sd khac nhau, ta si dung cac ky hieu khacnhau: f, g, cp,

Dinh nghia: Mien gid tri (MGT) cua mdt ham sd f la tap hgptdt ca cac sd thuc la gia tri cua ham sd do tai ft nhdt mot diemthudc mi£n xac dinh cua no.

Mien gia tri cua ham sd f xac dinh tren mien X duoc ky hieu laf(X):

f(X)={yeR: 3xgX sao cho f(x) = y).

b Ham so dang bieu thftc

6 bac hoc pho thOng, ban da dugc lam quen veri cac bieu thucchua bieh sd, ti nhung bi£u thtre co mdt phep toan deh nhungbieu thuc co nhi£u phep to£n phdi hgp, chang han nhu

xn, Vx , ax, logax, sinx, cosx, tgx, cotgx, ax2+bx + c 3x-lax +bx + c, - - , log2 -,

px + q 02 5-x

: - ■ , I Trifdng Daih<?c Kinh teQuocdan9i

Ta goi mien xac dinh tit nhien cua mdt bi£u thuc f(x) Id tap hop

tat ca cac sd thuc ma khi gan cho x thi bie’u thuc do co nghTa.Mdi bieu thuc f(x) la mdt ham sd xac dinh trdn mdt tap con Xbat ky cua MXD tu nhien cua no: mdi sd thuc xoeX dirge dattuong ling voi gia tri tfnh toan cua bi£u thuc do khi gan x = x0.

• Ve nguydn tac, MXD cua mdt ham sd la mdt tap sd thuc chotruce, con bieu thirc giu vai trd quy tac tuong ung f trong dinhnghia ham sd Do do, khi mdt ham sd xdc dinh trdn tap X cz Rduoc cho bang mot bieu thuc f(x), tap X cd the chi la mdt tapcon nao do cua MXD tu nhien cua bieu thuc do Tuy nhien,

10 TrtfdngDai hpcKinh teQuocdan

Chuang 1: Ham so va gicfi han

trong toan hoc nhibu khi ngubi ta cho trubc mdt bieu thuc f(x)va xet bieu thiic do nhu mdt ham sb Trong trubng hop nay taddng nhbt MXD cua ham sb voi MXD tir nhien cua bibu thircf(x).

• Mot ham sb co thb dirge cho dudd dang ph&n gia MXD thanhcac Up con rod nhau va tren mbi tap con do quy tac xac dinh giatri tuong ung cua ham sb tai mbi dibm dugc bieu dibn bang mdtbibu thurc ridng.

Vi du:

f(x) =

x2 +1 khi x > 0,l-2x khi x < 0.

la mdt ham sb xac dinh trdn R: trong khoang [0; +oo) gia tri cuaham s6 tai mbi diem x dugc tmh theo edng thirc f(x) = x2 + 1,con trong khoang (-oo; 0) gia tri cua ham so tai mbi diem x dugctmh theo edng thuc f(x) = 1 - 2x.

c Quan he ham so giita cac bien so

Trong cdc linh vuc khoa hoc ngubi ta phAn tfch quy lu£t thay doigia tri cua cdc dai lugng do dugc bang sb dudd dang cac bibn s6co quan hd vui nhau: su thay dbi gia tri cua bien so nay keo theosu thay dbi gia tri cua bibn s6 kia theo mdt quy luat nh£t dinh.Chang han, trong kinh te chung ta thby khi gia hang hoa thaydoi thi lugng hang hoa ma ngudd san xuat muon ban ra thitrubng va lugng hang hoa ma ngubi mua bang long mua cungthay dbi theo; khi thu nh|p cua cac hd gia dinh thay doi thilugng tidu dtmg cua ho cung thay dbi v v Su phu thudc cuamdt bibn sb nay vao mdt bibn sb khac thubng dugc bibu dibndudd dang ham sb.

Cho hai bibn sb x va y vdd mibn bibn thidn U cac tap hgp sb thucX va Y, trong do bibn x cd thb nh&n gia tri tu^ y trong mien bienthibn X cua nd Ta goi x U bien doc lap, hay doi so.

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, Trirdng Daihgc KinhteQuocdan11

IOAN CAO cXp CHO CAC NhA K1NH tg 1

Dinhnghia: Ta noi bien so y phu thudc ham so vao bien sox,

hay bien so y la ham so cua bien sd' x, khi vi chi khi tdn tai mdtquy tic hoac quy luat f sao cho mdi gia tri cua bidn sd x trongmien bien thidn X cua no duoc dat tuong ung vdi mdt va chi mdtgia tri cua bidn sd y.

Theo dinh nghia thi quy tic f chinh la mdt ham sd xac dinh trdnmien bien thidn X cua bien x va gia tri cua ham sd f tai diem xchinh la gia tri tuong ung cua bidn sd y:

xH>y = f(x).

De noi mdt each khai quat rang y la ham sd cua x (y phu thudcham sd vao x) ta co the viet: y = y(x).

Khi cho mdt ham sd f vdi MXD la tap hop X, cac each didn datsau day co nghia nhu nhau:

• Cho ham sd f xac dinh tren tap X (X la mdt tap sd chotruoc);

• Cho ham sd f(x), xgX;• Cho ham soy = f(x), xeX.

Chu y ring khi viet ham sd dudi dang y = f(x), cac ky hieu x vay chi mang y nghia hinh thuc, diing de gpi tdn cac bidn sd Mdtham sd duoc xac dinh boi hai ydu td: midn xac dinh X (mienbien thidn cua bien doc lap x) va quy tac f cho phep ta xac dinhduoc gia tri cua ham sd tai mdi diem xeX Chang han, dudi gidcdd toan hoc ta khdng phan bidt cac ham sd y = x2 va v = u2 khimien bidn thien cua x va mi£n bidn thidn cua u thing nhau.

ID DO THICIJAhAmSO'

Quan hd ham sd y = f(x) lidn kdt cac cap sd thuc (x0, yo), trongdd xo la mdt sd bit ky thudc mien xdc dinh X cua him sd va

Chuang 1: Ham so va gidi han

Vide lap dd thi cua mdt ham s6 f vdi mien xac dinh la mot

khoang sd thuc thuerng duoc thuc hien theo trinh tu nhu sau:

• Lay cac sd xp x2, , xn tit MXD cua ham sd (cang gan nhaucang tdt).

• Tfnh cac gia tri tuong ung cua ham sd tai cac diem do:yi = f(xi),y2=f(x2), ,y„=f(x„).

• Dinh vi cac die’m M/x/, y,), M2(x2, y2), , MnCx,,, y„).

• Ndi cac didm Mt, M„ ta duoc hinh anh dd thi cuaham sd.

Phuong phap dd thi khdng phai ia phuong phap chinh xac Tuynhidn, ngubi ta thubng su dung dd thi de minh hoa bang hinh

1 vANUAw UAH vl ivUAw NriA KINH 1

anh cac dac trimg cor ban cua sir phu thuoc ham sd giua cac bidnsd Nhin vao dd thi ta dd dang quan sat xu hirdmg bidn thidn cuaham so khi bien dOc lap thay ddi gia tri.

IV KHAI NlfiM HAM NGUOC

Xet mot ham so y = f(x) vdri midn xac dinh X va midn gia triY = f(X) Neu vdri mdi gia tri yoeY chi tdn tai duy nhdt mOtgia tri x^X sao cho f(x0) = y0, tire la phiromg trinh f(x) = y0 comot nghi£m duy nhat x0 trong mien X, thi

y = f(x) <=> x = f“l(y) (xeX, yeY),

trong do ky hieu x0 = f -,(y0) chi nghiem duy nhat cua phiromgtrinh f(x) = y0 nhir da noi d tren.

Nhir vay, trong tnrong hop nay quan hd ham sd y = f(x) bidudidn sir phu thuoc cua y vao x co thd dao nguqc de bieu didn sirphu thudc cua x vao y thong qua ham so x = f -1(y).

Dinh nghTa: Vdri gia thidt va quy udre ve ky hidu ndu tren, ta goiham sox = f-I(y) la ham ngitcrc cua ham sd y = f(x) Ndi each

khac, ham so f~' (xac dinh tren midn Y = f(X)) la ham ngitac

TV TV

Chuang 1i Ham $6ya

• H&m sd y = sinx vdi mi£n xac dinh X = cd hamngugc la h^m sd x = arcsiny (- 1 < y < 1), trong do ky hi£uarcsiny0 chi nghiSm duy nhdt cua phucmg trinh sinx = y0 trong

khoang < x<-

• Ham sd y = cosx vdi mien xac dinh X = [ 0; tc ] cd hamngirgc la ham sd x = arccosy (- 1 < y < 1), trong do ky hieuarccosy0 chi nghiem duy nhdt cua phucmg trinh cosx = y0 trongkhoang 0 < x < 7t.

• Ham sd y = tgx vdi mien xac dinh X = cd hamngirgc la ham sd x = arctgy (ye R), trong do ky hieu arctgy0 chinghidm duy nhdt cua phucmg trinh tgx = y0 trong khoang

-< X <—.

■ 2 2

• Ham sd y = cotgx vdi mi£n xac dinh X = (0; n ) cd ham

ngirgc la ham sd x = arccotgy (yeR), trong do ky hieuarccotgy0 chi nghiem duy nhat cua phucmg trinh cotgx = y0 trongkhoang 0 < x < it.

thi, bdi vi y = f(x) va x = r’(y) la cac phucmg trinh tucmg duomg.Tuy nhifcn, trong toan hoc ngudi ta thudng dung ky hi£u x de chibidn ddc lap va ky hi6u y de chi bidn phu thudc, do do thay choeach vidt ham ngugc dudi dang x = f-1(y) ngudi ta cd the trao kyhieu bidn sd va vidt ham ngugc cua ham sd y = f(x) dudi dangy = f -1(x) Chang han, ta cd thd ndi: ham sd y = logax la hamngugc cua hilm sd y = ax, hay dom gian hem: logax la ham ngugccua ham sd ax.

toAn GAO cXpchocAcnhakinht£

Do trao ky hieu bien sd' n6n didm M(x, y) thudc d8 thi y = f“’(x)khi va chi khi di8m M’(y, x) thudc d8 thi y = f(x) Trdn matphang toa do hai diem M(x, y) va M’(y, x) db'i xumg nhau quadudmg phan giac thii nhat Nhu vSy, ndu bidu didn hai d8 thiy = f(x) va y = r’(x) tren cung mdt hd toa dd true chuin thichung doi xiing nhau qua duong thing y = x (dudmg phan giaccua goc phan tu thur nhat) Chang han, hai dudmg cong y = ax vdy = logax co dang nhu hinh ve dudi day.

V.MOT • •SODAC TRUNG HAMSO

a Ham so dm dieu9

Dinh nghTa:Ta noi rang ham s6 y = f(x) dm dieu tang (dm

thudc X, hieu so f(x2) - f(Xj) ludn ciing d&i (trai da'u) vdi x2 - xPNoi each khac:

• Ham so f(x) la ham dem dieu tang tren midn X nduX1<x2 => f(xt)<f(x2) (Vx1,x2GX).• Ham so f(x) la ham dem didu giam trdn mi8n X ndu

16 Trtrdng Dai hocKinhto Quoc dah > h

Xj<x2 => f(x,)>f(x2) (Vx1,x2GX).

Ham so don dieu tang (don dieu giam) cdn diroc goi la ham so

dong bien (ham so nghich bien).Vi div.

• Ham so f(x) = x2 la ham don dieu tang tren khoang [0; +oo)va don dieu giam tren khoang (-00 ; 0]:

Vxj, x2 g[0; +00): X| < x2 => x,2< x22;VX|, x2 e(- 00; 0]: x, < x2 => x,2>x22.

• Ham so f(x) = — dan dieu giam trong khoang (0; +00):x

1 1Vx., x2 g(0; +00): x < x2 => — > — '

X| x2

Neu quan sat do thi cua ham so theo hudng tir trai sang phai thi

do thi cua ham so dan dieu tang cd dang doc len va do thi ciiaham so dan dieu giam co dang doc xuong.

TrudngDai h?cKinh teQuoc dan17 i

IOAN CAO CAP CHO CAC NHA KI NH Tg •

duoc hieu theo nghia dam dieu ngat Ta co the mo rong khainiem ham don dieu nhtf sau:• •

• Ham so f(x) duoc goi la ham dem dieu tang theo nghia rong

tren mien X neu

Xj<x2 => f(x,)< f(x2) (VxHx2eX).

• Ham so f(x) duoc goi la ham dem dieu giam theo nghia rong

tren mien X neu

Xj<x2 => f(Xj)> f(x2) (Vx,,x2€X).

b Ham so bi chan

Dinh nghia:

• Ham so f(x) duoc goi la ham bi chan trong mot mien X neu

gia tri cua ham so chi thay doi trong pham vi mot tap con cuamot khoang so huu han khi x bien thien tren mien X, tire la tontai cac hang so m va M sao cho:

m < f(x) < M Vx g X.

• Ham so f(x) duoc goi la ham bi chan tren trong mot mien Xneu ton tai hang so M sao cho:

f(x) < M Vx gX.

Hang so M duoc goi la can tren cua ham so f(x) trong mien X.

• Ham so f(x) duoc goi la ham bi chan diced trong mot mien X

neu ton tai hang so m sao cho:

f(x) > m Vx g X.

Hang so m duoc goi la can diced cua ham so f(x) trong mien X.Chu y rang tinh bi chan bao ham cd bi chan tren va bi chan

va chi khi ton tai hang so K > 0 sao cho:

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18Trtfdng Dai hoc Kinh te Qudc dari |

C/?agng 7.-/7azn so vagjd?ftan

Dd thi cua h&m sd chan va h&m sd le co tmh chat ddi xung: do

thi cua ham so chan nhan true tung lam true doi xicngf con dothi cua ham sd'le nhan gd'c toq dd lam tarn do'i xi(ng.

7 toAn CAO cXpchocAc NHA KINH ig "' "' " '"'jT j

d Ham sd tudn hoan

Dinh nghia: Ham sd f(x) xac dinh trfin midn X dirge goi Id ham

sin(x + 2k) = sinx; cos(x + 2k) = cosx, Vxg R.

T = k:

tg(x + 7t) = tgx, Vx * — + kre; cotg(x + k) = cotgx, Vx kK

CAPDOIV6l HAM SO

a Cac ham sd sot cap cot ban

Thdng thirdng mdt ham sd y = f(x) dirge cho dirdi dang mdt biduthifc hiru han Cac bidu thde hdu han dirgc<hgp thanh td cdc biduthde ham sd ca ban.

Cac ham sd sau day dirge goi la cac ham sdset cap cctbdn:1 f(x) = C (ham sd nhan gid tri khdng ddi C vdi moi x).2 Ham sd luy thira: f(x) = xa (a = const.).

y" .J -te-aer - - - y I

20 JTrtrdng Dai hpc Kinh te Quoc dan

- Chactngl :Hamsd va gicihan

3 Ham s6 mu: f(x) = a (a>Ovaa?* l).*

4 Ham sd logarit: f(x) = logax (a>Ovaa^l).

5 Cac ham sd lirgng giac:

f(x) = sinx, f(x) = cosx, f(x) = tgx, f(x) = cotgx.6 Cac ham lirgng giac ngirgc:

f(x) = arcsinx, f(x) = arccosx, f(x) = arctgx, f(x) = arccotgx.

bang radian.

b Cac phep toan sa cap ddi vdi ham so

Cac phep toan sa cap doi voi ham sd bao gdm: phep cong, pheptrir, phep nh&n, phep chia va phep hop ham.

• Cac phep toan cdng, trir, nhAn, chia ddi vdi cac bie’u thuc

ham s6 dugc thgc hidn gidng nhir ddi vdi cac bieu thvrc dai sd.

Ndu f(x) va g(x) 1& cac ham sd cho diroi dang bieu thiic thi cacbidu thirc

f(x) + g(x), f(x) - g(x), f(x)g(x),

dirge goi tuong ling la tdng, hidu, tfch, thirong cua f(x) va g(x).Cac ham sd nay dat tirong ling mdi gia tri cua bidh ddc lap x vditdng, hidu, tfch, thirong cac gia tri cua cac ham sd f va g tai diemx:

f(x) + g(x): x f-> y = f(x) + g(x)f(x) - g(x): x I-4 y = f(x)-g(x)f(x).g(x) : x f■> y = f(x).g(x)

g(x) ’ X

fwy g(x)

■ ■— ■' ■- I, j|

: TnrdngDaihoc Kinh teQuoc dan 215

Vi du:

Ham sd y = x + sinx la tdng cua hai ham sd:f(x) = x, g(x) = sinx.Ham sd y = x3log3x la tich cua hai ham sd

<p f

x h> u = <p(x) y = f[cp(x)J = g(x).

Ham sd y = g(x) = f[cp(x)] dat tuomg ung mdi gia tri cua bidn sdx vdi mot gia tri duy nhdt cua bidn y theo quy tac ndu tren dirgegoi la ham hap cua cac ham sd y = f(u) va u = cp(x) Ham hgpcon dirge goi la ham kep Bo qua vai tro hinh thtfc cua cac kyhieu bidn sd ta co thd ndi: g(x) = f[q>(x)J la ham hgp cua hai hamsd f(x) va tp(x).

Vi du:

Ham sd y = sin5x la ham hgp cua hai ham sd y = u5 vd u = sinx Tacung cd the ndi: g(x) = sin5x la ham hgp cua hai ham sd f(x) = x5 vdcp(x) = sinx.

c Cac ham so sa cap

Ta goi ham sd sa cap la ham sd dirge cho dtfdi dang mdt bi£u

TrUdng Daihgc Kinh te Quoc dan

vn m6tsomohinhhamso TRONG PHANtick

a Ham cung va helm cau

Khi phan tich thi trircmg hang hoa va dich vu, cac nha kinh sirdung khai nifim ham cung (supply function) va ham can(demand function) de bi£u di£n sir phu thudc cua I item g cung va

Ham cung va ham cdu co dang:

Ham cung: Qs = S(p),Ham c<lu : Qd = D(p),

thirc huu han, tire la mOt bi£u thirc dirge hop thanh tir cac ham sdso cap co ban ndi tren thdng qua mdt so huu han cac phep toanso cap ddi vdi ham sd.

Pham vi cua tap hop cac ham so cap kha rdng Trong kinh te hocnguofi ta thuefng hay sir dung cac dang ham so sau:

• Ham sd f(x) = axa (a va a la cac hang sd).• Ham sd mu va ham sd logarit:

f(x) = ax, f(x) = logax (a > 0 va a * 1).• Ham da thirc, hay ham nguyeni

• Ham phan thirc, hay ham huu ty:

„ pWW=QWtrong do P(x) va Q(x) la cac da thirc.

Cdluuny j+ rtarn SO va y/v/ nan

■ TriJfcina Dai hoc Kinh te

•T:';!- •J1’-;-:* h M si • < 1 • ■ ■ M ’! MUQc aan zdi

r TOAN CAO cXp'CHO CAC NHA KINH Itrong do: p la gia hang hoa; Qs la luomg cung (quantitysupplied), trie la luomg h&ng hoa m& ngudri ban bang Idng ban;Qd la liromg cdu (quantity demanded), tuc 1& liromg hang hoa mangudri mua bang long mua Trong md hinh ph&n tich thi trudmgmot loai hang hoa, luomg cung cua thi trudmg la tdng luomg cungciia tat ca cac nha san xu<ft va luomg c£u cua thi trudmg la tdngluong c£u cua tat ca nhung ngudri tidu dhng.

Tat nhien, luomg cung va luomg c&i hang hoa khdng chi phu thudcvao gia cua hang hoa do, ma con chiu anh hudmg cua nhieu ydu td'khac, chang han nhu thu nhap va gia cua cac hang hoa lidn quan.Khi xem xet cac md hinh ham cung va ham c&i dr dang neu trdnngudri ta gia thidt rang cac yen to khac khdng thay doi Quy hi&t

thi trudmg trong kinh td hoc noi rang, ddi vdri cac hang hoa thdngthuomg, ham cung la ham dem dieu tang, con ham cau Id ham dem

khi gia hang hoa tang len thi ngudri ban se mudn ban nhidu hem v&ngucri mua se mua ft di Cac nha kinh td goi d6 thi cua ham cung

va ham cdu la dieting cung va dieting cau Giao difim cua duomgcung va duemg cau duoc goi la diem can bang cua thi trudmg: 6

mire gia can bang p ta cd Qs= Qd= Q, tuc la ngudri ban ban hdtva ngucri mua mua du, thi trudmg khdng co hidn tuemg du thftahoac khan hidm hang hoa.

hoanh de bieu didn luomg Q va true tung de bid’u didn gia p.Cach bieu didn nhu v|y tuemg ling vdri vide dao nguqc ham cungva ham cdu dr dang noi trdn Trong kinh td hoc nhidu khi ngudrita van goi ham ngucrc cua ham Q = S(p) la ham cung va hamnguoc cua ham Qj = D(p) la ham cdu:

Qs = S(p)«> p = S-’(Qs),Qd = D(p)« p = D-(Qd).

:24 TrirdngDai hoc Kinh te Quoc dan

D6 thi cua ham cung va ham cdu (dircfng cung va dircmg c£u) codang nhir hinh ve.

i Chuang 1: Harn sava gidi han \

Di£m can bang la diem( Q, p ), trong do Q la lirgng can bangva p la gia can bang.

b Ham san xuat ngan han

Cdc nha kinh hoc s£r dung khai ni6m ham san xuat de md tasir phu thudc cua san lirgng hang hoa (t6ng s6' lirgng san phamhi6n vat) cua m6t nha san xuat vao cac y£u td ddu vao, goi la

cdc yeu to san xuat, nhir vdn va lao dong v v

Trong kinh te hoc khai ni6m ngan han va dai han khong dirgexac dinh bang m6t khoang thcri gian cu th6, ma dirge hieu theonghia nhir sau:

Ngan han la khoang th&i gian md it nhd't mot trong cdc yeu td'san xuat khong the thay doi Dai han la khoang thcri gian md tatcd cdc yeu td'san xuat co the thay doi.

Khi phan tich san xuat, ngirdi ta thirbng quan tarn den hai y£u tdsan xuat quan trong la vdn (Capital) va lao dong (Labor), dirgeky hi6u tirong ung ia K va LI

Trirdng Dai h9c KinhteQuoc dan 25

ItoAncapcXpchocAcnha k inht£

Trong ngin han thi K khdng thay ddi, do do ham san xu£t nganhan co dang:

Q = f(L),

trong do L la lirgng lao ddng dirge sir dung va Q la mire sanlirgng tirorng ung Chu y rang khi xet ham san xu£t, san lugng Qva cac yeu td san xuat K, L dirge do theo ludng (flow), tire la dodinh ky (hang ngay, hang tuin, hang thang, hang nam v.v ).

c Ham doanh thu, ham chi phi va ham lai nhuqn

Tong doanh thu (total revenue), tong chi phi (total cost) va t6ngIgi nhuan (total profit) cua nha san xu£t phu thudc vao san lirgnghang hoa Khi phan tich hoat dong san xuat, ciing vdri ham sanxuat cac nha kinh te hoc con sir dung cac ham sd':

doanh thu (ky hieu la TR) vao san lirgng (ky hidu la Q):TR = TR(Q).

Ching han, tong doanh thu ciia nha san xuat canh tranh la hambac nhat:

TR = p.Q,

trong do p la gia san phim tren thi tnromg Dd'i vdi nha san xu£tdoc quyen, tong doanh thu dirge xac dinh theo edng thiic:

TR = D"’(Q).Q,trong do p = D-1(Q) la ham ciu ngugc.

phi san xuat (ky hieu la TC) vao san lirgng (ky hidu 1& Q):TC = TC(Q).

■I I,.I I _ _ _ - -i -—■ , ITT

25 Tr hoc Kinh te Qu& ^an :

Chu'cmg1:Ham sovagidi han

• Ham leri nhuan la ham sd bi£u didn su phu thudc cua tdng loti

nhu^n (ky hidu la k) vao san lugng (ky hidu la Q):7C = 7t(Q).

Ham led nhu3n co thd dirge xac dinh thdng qua ham doanh thuva ham chi phi:

K = TR(Q) - TC(Q).

d Ham tieu dung va ham tiei kiem

Lirgng tidn ma ngirdri tieu diing danh de mua sam hang hoa vadich vu phu thudc vao thu nhap Cac nha kinh X.& si dung ham

(Consumption) vao bien thu nh|p Y (Income):C = f(Y).

Thdng thudmg, khi thu nhAp tang ngucri ta co xu huemg tidu diingnhidu horn, do do ham tidu diing la ham ddng bien.

Ham tiei kiem la ham sd bieu didn su phu thudc cua bidh tiet

kiem S (Saving) vao bidn thu nhap:S = S(Y).

BAI TAP

Cho ham sd: f(x) = Vx2 + x + 2 Hay tmh:f(l), f(-2), f(-a), f - ,f(a + b).2 Cho ham sd':

f(x) =

x2 + 1, khi x < 0,2x,khix>0.Hay tmh: f(-4) f(-3), f(-l), f(0), f(4), f(5).

TnjfdngDai hoc Kin hte Quoc dan 27r

toAncaocKpcho cAc nhA kinht£ •

4 Tim MXD cua cac ham sd:

b) y = lg(x2-9)d) y = log2log3log4Xy = V2x — x2

y = arcsin(l - x) + lg(lg x)3 Chung minh:

1, neu x la sdhiru tyf(x) 7

10 Chung minh rang neu ham sd f(x) don dieu tang (giam) trongkhoang X thi ham sd -f(x) don di6u giam (tang) trong khoang do.

12 Voi cac ham sd co cung MXD, hay chung minh:

a) Tdng cua hai ham sd chan la mdt ham sd chan;b) Tdng cua hai ham sd le la mdt ham sd le;

c) Tich cua hai ham sd chan la mdt ham sd chan;d) Tich cua hai ham sd le la mdt ham sd chan;

e) Tich cua mdt ham sd chan va mdt ham sd le la mdt ham sd le.

13 Goi f-1 la ham nguoc cua ham sd f Hay vidt bieu thirc f"1 (x)trong cac tnrdng hop sau:

a) f(x) = ax + b,-oo <x<+°o (a^O);

b) f(x) = x2 , 0 < x < +oo; <c) f(x) = x2 , -oo < x < 0;

d) f(x) = 1—— , x-l.* r-, 1 + X

14 Cho f(x) = x3 - x, g(x) = sin2x Hay Up cac bieu thtfc ham sd:a) f[g(x)] b) g[f(x)] c) f[f(x)] d) g[g(x)]

15 Hay Up bi£u thuc ham sd f(x), cho biet:a) f(x + 1) = x2 - 3x + 2

' f oAn 6Ad cXp CHO CAC NHA KINH

te-.-.—; - - —T — —— -; -;' ■- - -; - - ;——; i—. a

16 Bifit rang f(x) la mdt ham sd xac dinh va dan difiu giam trongkhoang U Chung minh rang nfiu ham sd <p(x) xac dinh, don difiu tang(giam) trong khoang X va (p(X) c U thi ham sd g(x) = f[cp (x)] dondifiu giam (tang) trong khoang X.

17 Hay difin thudt ngu thich hop vao chO ddu ba chdm trong cac cdudircri dAy:

f) Ham sd bieu difin anh huong cua thu nhdp ddi vdi luong tifiudiing duoc goi la

18 Cho bidt ham cung va ham cdu cua thi tnrcng m6t hdng hod nhusau:

Q, = 4p-1; Qd= 159-2p2.

a) Hay so sanh luong cung vdi luong cdu d cdc mile gid p = 7,p = 8,l.

b) Xac dinh gia edn bdng vd luong edn bdng cua thi tnrcng.

30 TrUdngDai hoc Kinhte Quoc dan

19 Mot doanh nghidp co h^m san xuS't nhu sau:q = iooVl7,

trong do L la luong str dung lao ddng va Q la luong san pham ddu ramdi tu^n.

a) Hay cho biet luong san pham dau ra mdi tuan khi doanh nghiepstr dung 64 don vi lao dOng mdi tuan va giu nguyen mile su dungcac yeu to dau vao khac.

b) Tai muc str dung 64 don vi lao dong mdi tuan, neu doanhnghiep them 1 don vi lao dong mdi tuan thi san luong dau ra mdituan tang bao nhieu (tinh xap xi den 1 chu so thap phan)?

20 Mot nha san xuat cd ham chi phi nhu sau:

TC = Q5 - 5Q2 + 20Q + 9.

a) Hay tfnh tdng chi phi san xuat tai cac muc san luong Q = 1,Q = 2 va Q = 10.

b) Cho biet chi phi cd dinh va ham chi phi kha bien.

21 Vdi ham chi phi cho O bai tap 20, hay lap ham loi nhuan cua nhasan xuat trong cac trudng hop sau:

Mot each don gian ta goi day so vd han la mot day liet ke cac so

thuc theo thii tu nhu sau:

x„ x2, x3, , xn, (2.1)

Trirdng Dai ho| Kinhte Quoc dan31!

I?- 'TOAN CAO CAP CHO CAC NHA"KINK Tg?5':''' ■ j

So xn 6 vi tri thu n trong day liet ke (2.1) dupe goi la so hang thit

xn= f(n) vdi doi so n bien thien tren tap hop tat ca cac so tu nhienN= {1, 2, 3, 4, Ngupc lai, moi ham so doi so tu nhienxn= f(n) cho tuong ung mot day so viet dirdi dang (2.1) Ta goiham so xn= f(n) la so hang tong quat cua day so (2.1) Do co sirtirong ung ndi tren, ta dong nhat day so (2.1) vdi ham so f(n) (sohang tong quat cua no) Thay cho each viet khai then duoi dang(2.1) ta co the goi don gian: day sox„=f(n), hoac day so xn.

1,4, 9, 16, , n2,

day so voi so hang tong quat xn = a + d(n - 1):

a, a + d, a + 2d, , a + d(n - 1), (2.2)Day so (2.2) co tinh chat sau:

xn = a + d(n - 1) = a + (n - 2)d + d = xn , + d, Vn > 2.

Dieu nay co nghTa la mOi so hang cua day so (2.2), bat dau tir sohang thu hai, bang so hang dung truoc no cong vdi mot hang sod Day so (2.2) duoc goi la cap so epng va hang so d dupe goi la

cong sai cua no.

z n —

tong quat xn = aq :

a, aq, aq2, aq3, , aqn’ (2.3)Mdi so hang cua day so (2.3), bat ddu tir so hang thur hai, bangso hang dung trudc no nhan voi mot hang so q:

x„ = aq"'' = (aq""2)q = x„-,q, Vn > 2.

32 Trudng Dai hoc Kinh te Quoc dan

I

~ Chu^ngljHamsd^^ffK# frgn ;

Day s6 (2.3) dirge goi U cap so nhan va hang sd q dirge goi la

edng boi cua nd.

II.Gl6l HAN CUA DAY SO

a Khai niem day so hoi tu

Khai nidm gidi han trong toan hoc bieu didn xu hirdng bienthien cua mdt bidn s6 ngay cang lien gdn den mdt sd nao dd Tir“tien g&i” bao ham khai niem ve khoang each Nhir ta da biet,khoang each giua hai s6 a va b dirge hieu theo nghia khoangeach giua hai di£m tircrng ung tren true sd va dirge xac dinh theoedng thtfc: d(a, b) = | a - b |.

6 a b" x

Gidi han cua day sd xn bieu didn xu hirdng bidn thidn cua xn khin Idn vd han.

Dinh nghia: Ta ndi rang day so xn co gidi han a, hay xn hoi tu

y bang each Dy n du Idn, tue la vdi moi sd e > 0 be tuy y, baogid cung cd th£ tim dugc tuefng ling mdt sd' tu nhien Hq du Ionsao cho

I x„ - a | < e (2.4)bat d£u tir khi n > r^.

D£ ndi day sd xn hdi tu den a ta diing ky hidu:

lim x_ = a, hoac x-> a khi n —> +oo.

Bdt dang thde (2.4) cd th£ viet dudi dang:

—s<xn — a<8 <=> a-8<xn<a + e <=>xne(a-s; a + e).Khoang Ve(a) = (a - s; a + e) dugc goi la Ian can ban kinh 8

Trurdng Daihsc Kinh te Quocdan

-

Day so x„ hdi til den diem a khi vd chi khi moi Ian can ban kinhnho tuy y cua diem a deu chita tat cdc cdc sd' hang cua day sodo bat dan tit mot chd nao do trd di.

D6 chung minh day sd xn hdi tu ddn didm a, theo dinh nghia, taphai chi ra sd tu nhidn tirotag ting vdi m6i sd 8 > 0 sao cho batdang thtrc (2.4) thoa man vdi moi n > Hq.

|xn- c| = | c - c | = 0 < e, Vs > 0 va Vn.V|y, theo dinh nghia: lim c = c

, 2n +1

nTa co:

Khoang each giua x,, va sd 2 cd thd thu hep tuydn< 0,1 khi n > 10,

dn< 0,01 khi n > 100,dn< 0,001 khi n>1000.

Vdi 8 la mdt sd ductag bdt ky, dn= |xn- 2 | = — < 8 khin

n > n0 = [e-1] (ky hidu [x] chi ph£n nguydn cua sd thtfc x).Theo dinh nghia ta cd:

Chuong 1: Ham so va gidi han

hang sd duong cho trucrc Ta co:

i -i

lx - 0| = —< e <=> n > — <=> n > 8 k n e

Theo m6i sd 8 > 0 ta chon n0 la ph£n nguySn cua sd s k D6 tosd tu nhifin thoa man didu ki&n |xn— 0| < 8 khi n > n^ Nhu v&y,theo dinh nghia ta c6:

011111 IV：^461 637，6X” 1161 (11 (1血 11161 86 3, 1130 (16 1(111 膝血场］^61 01016〉0 111611 1611(211 111002 血8 01 但 8^ 1111111德II 00 (10160830 0110 顷 山化

011111 0^1113：%1161 1^05 明〉X，00 腔花【I即 V。【I即 0^11 X” 06^13 III 111X^1 ^61 1而：11/ V 炒］11 册 160, 1^0 1 祝 ^61 11101，6 8〉016x1 1117 勺,8但 01102^6 I槌 1101(11X0^0 0161 8^ 中 口陌曲 00(111160830 0110

斑I 描11 ［"炒 1II〉队.^461 86 06 8面 11311 V。1130 000 ^典^

Ndu day sd xn cd gidi han vd han thi ta vidt:

lim xn = oo hoac xn—> oo khi n —> 4-oo.

Chu'ong 1: Hamva gidi n an

Ndu day sd xn cd gidi han vd han va xac dinh ddu, tire 1& xn > 0hoac xn < 0 bat dau tir mdt chd nao do trd di, thi ta vidt tuongling;

lim x_ = 4-oo, hoac lim xn = -oo.

n->+<» • n->+co

dd chi cac d£u mut cua true sd, va ky hidu oo cd nghia la -oohoac 4-oo Cac ky hidu nay chi mang y nghia hinh thuc de didndat y nidm vd han, khdng nam trong pham vi hd thdng s6 thuc.Do do, ndu khdng cd quy udc bd sung thi ta khdng thd ap dungcac phep toan sd hoc ddi vdi —oo v& 4-oo.

do A * 0 va k > 0 la cac hang sd cho trudc Vdi E la mdt sdduong ba't ky, ta cd:

Tuy theo da'u cua A ta c6 thd vidt:

lim Ank = 4-oo ndu A > 0; lim Ank = -oo ndu A < 0.

n-Tri/6’ng Da h9c Ki nK ta Quocdan 3Z

~ toAncaocApcho CAC NHA KINH

Tit dinh nghia gidi han va dinh nghia VCB suy ra:

Dinhly: Day sd xn hdi tu den didm a khi va chi khi day sda,, = Xn - a la mdt VCB Noi each khac, day sd xn hdi tu ddndidm a khi va chi khi nd bidu didn duorc dudti dang xn = a + an,trong do an la m6t VCB.

b Mot so tinh chat dan gidn cua yd ciing be

1 Neu an va pn la cac VCB thi otn ± pn cung la VCB.

la sd Idn han trong hai sd nb n2, ta c6

I ctn ± pn I < I (Xh | + | pn | < e, Vn > n0.Didu nay chdng to a,, ± pn la VCB.

2 Neu a,, la VCB va un bi chan thi anUn cung la VCB.

That v|y, day sd u,, bi chan cd nghia la tdn tai hang sd K > 0 sao cho

nhidn n0 sao cho | a,, | < e/K Vn > n<), tir day suy ra:

I unan | = | un | | an | < K^- = e, Vn > n0.

Didu nay chtrng td aoun la VCB.

I -.'l.'JiH - I IIIJJ J J J II ' J>J"'■ 1 'WI>IRI,I| , I,

IV CAC DINH• •LVCO BAN Vfs GlCl HAN

a, Cac tinh chat cot ban cua day so hoi tu

Dinhly1: Ndu day sd xn h6i tu ddn s6 a thi no khOng the h6i tuden m6t sd b a N6i each khac, giai han cua mot day so hoi tu

la mot sd'thitc duy nhdt.Chitngminh: Vdi b* ata cd:

|xn- a| + |xn- b| > |(xn- a) - (xn - b)| = |a - b|.

thcri thoa man:

|xn-a|<E, k-b^s.

N6u lim X = a thi khi n du Idn ta ludn c6 k - a| < s, do dd |xn - b| > s.

Di£u nhy chiing td s6 b khdng phai ia gidi han cua day s6 x^.

Dinh ly 2: Ndu day sd xn h6i tu thi n6 bi chan, tire la tdn tai cachang sd A, B sao cho A< xn < B vdi moi ne N.

Chi'cng minhz Gia str day sd xn hdi tu ddn sd a Vdi e0 ia mdt sd duong chotrade ta tim dirge sd tu nhifin n0 sao cho

a - e0 < xn < a + e0, Vn > n0.

Dat A = min{Xj,, XIlo,a-Eo},B = max{x1,Xn<j ,a + Eo), tac6A < xn < B vdi moi ne N.

He qua: Ndu an Pn la cac VCB thi cung la VCB.

That v£y, tit dinh ly 2 suy ra moi VCB £n d£u bi chan Do d6, ndu a,, vh £n iac£c VCB thi otnpn cung ia VCB (ta da chdng minh rang tich cua mdt VCB vhmdt day sd bi chan ia mdt VCB).

Dinh ly 3: Ndu lim xn = a a > p (a < q) thi xn > p (xn < q)bat d£u tir mdt chd n&o do tier di Dae biet, ndu a > 0 (a < 0) thixn > 0 (xn < 0) khi n du 1dm.

toAncaocXpcho cAc nhA KINH if '

Chung minh: Do lim xB = a nfin theo sd e > 0 bfl't ky ta tim dirge sd tg

n-nhiSn rig sao cho bdt ddu tir khi n > n0:

dirge sd tg nhi6n n0 sao cho:

xn<P<yn»

Vn>no-Di£u n&y mAu thuSn vdi gia thidt x„ > yB, Vne N MSu thuln n&y bdc bo giathidt a < b, do d6 a > b (didu phai chting minh).

Chu y: Khi xn > yn voi moi n e N ta cung chi c6 th6 kdt luAn

diroc rang lim xn > lim y_, bod vi c6 kha nang xay ra tnrcmg