Hiểu về khái niệm 'giới hạn' trong toán học (kỳ 1) _ Math2IT.pdf

1/15/2015 Hiuvkhinim'giihn'trongtonhc(k1)|Math2IT

http://math2it.blogspot.com/2014/09/hieuvekhainiemgioihantrongtoan.html 1/9

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Dinh Anh Thi | 10.9.14 | Hiu ton hc | Ton s cp | 7Like

Hiu v khi nim 'gii hn' trong ton hc (k 1)

Trong Ton hc, khi nim "gii hn" c s dng ch gi tr m mt hm s hoc mt dys tin gn n khi bin s tng ng tin gn n mt gi tr no . Tuy nhin khi nim trutng ny kh kh c th tip thu. Bi vit hy vng phn no gip bn hiu v sao li c khinim ny trong ton hc v mi lin h thc t ca n ra sao.

Ta bt u i vo tm hiu khi nim gii hn (limit) trong ton hc thng qua cc nghch l nt thc t cuc sng.

Nghch l Znon

Trit gia Hy Lp, Znon x Elea (490 BC - 430 BC)

c th i vo hiu r ti sao li c khi nim "gii hn" trong ton hc (limit hay vit tt l lim)

TRANG CH LATEX TON LINUX OS X CNTT TRI THC KHC MATH2IT tm



c th i vo hiu r ti sao li c khi nim "gii hn" trong ton hc (limit hay vit tt l lim)ta hy cng nhau tm hiu hai cu chuyn nghch l ca Zenon. Cu chuyn u tin ni v vthn Achilles (ting Vit thng gi l Asin) - mnh danh l v thn c th di chuyn vi tc nhanh nht m c th gian khng ai bng trong thn thoi Hy Lp. Cu chuyn th hai ni vcch mt mi tn i n ch nhng khng bao gi ti c.

Cu chuyn 1 - Achilles chy ua cng raMt ngy n, thn Achilles chy thi vi mt con ra. Do c mnh danh l thn v tc nnAchilles nhng ra mt on, Achilles ti , ra ti . C hai xut pht cng mt lc, theocng mt hng v nhim v ca thn Achilles l phi ui kp con ra.

Ch trong nhy mt, khng my kh khn, Achilles n c . Th nhng d ra chy chmth vn tc ca n vn ln hn 0 v n i n c . Tip tc, Achilles ui n th ra n

, Achilles ui n th ra n .

C tip tc nh th, cc im ny lun lun tn ti v nh th th Achilles, mt v thn v tc li khng ui kp mt con ra. iu ny l v l theo l thng tnh, nhng hon ton khng cg mu thun trong lp lun trn, vy iu g ang din ra?

Cu chuyn 2 - Mi tn khng bao gi trng chTa bn mt mi tn n bia. Nu t v tr ch xut pht tn l , bia l .

Mi tn mun n c bia th phi i qua trung im ca . T mun n c phi qua trung im ca . T mun n phi qua trung im ca

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C tip tc nh vy, mi tn phi ln lt qua trung im ca cc on thng chia nh, m sim trong mt on thng l v hn ngha l mi tn phi i qua v hn im, ng ngha vivic khng bao gi n c ch.

Nhn xtTa bt u tm hiu s hnh thnh nn khi nim gii hn t hai cu chuyn th v trn, bt uvi hai nhn xt:

Th nht, hai cu chuyn trn u c th xy ra trong thc t v chnh iu lm chochng tr thnh nghch l ni ting. Hon ton khng c th thut hay mo vt g y nhcc cu vui m bn hay gp phi.Th hai, cng t thc t, Achiles khng th no khng th chy bng ra, v kt qu l anhta s chy c n ng ch con ra ang ng. Cng nh mi tn khng c chuyn baymi m khng ghim vo tng. Thc t chc chn n s n c ch.

Phi tha nhn rng, gii hn khng phi pht minh t trn tri ri xung m con do con ngit to ra em li nhng iu vui th mang tnh cht gii tr. Con ngi gp phn "phthin" ra nh ngha ny.

Cng nh ton hc, mi nh ngha, nh l, cng thc, u c ngha ca n v u n tcuc sng. Ci khc m ti mun khng nh y chnh l con ngi bt u nhn thy s tnti ca ton hc, m t nhng th m h thy c, tm ra nhng quy lut c bn nht ri t dng chng d on "tng lai", d on ra c nhng quy lut m chnh chng ta cngcha pht hin c. Chnh nhng kt qu d on n qu xa l vi chng ta nn lm chochng ta ngh rng ton hc lm ra nhng iu v ngha, l thuyt v so iu. Ch n khinhng d on tr nn hu dng v lm cho chng ta quen thuc, ta bt u chp nhn vnhn ra ", c l n cng c cht hu ch". chnh l ton hc. Mt cht tn mn.

Ta xt v d v mi tn, nu thi im bt u bn, khong cch gia mi tn v bia l .

tng c bn v khi nim gii hn theo ngha ton hc t hai v d trn l g? Tnh kh

thi v thc t ca n ra sao?

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Ln 1: mi tn n c trung im ca , khi y khong cch l



Ln th : mi tn n c trung im ca , khong cch l

n y bn ngh mi tn c dng li khng? Xin tha l khng, n vn c tip tc bay i, baymi. y ta ang xt n thc t l khong cch gia ngi cung th v bia khng qu xa, vnu n qu xa, lc bn khng th do trng lc, mi tn s ri gia ng. Thc t l mi tnvn ghim thng vo bia, khi y c ngha n c qua trung im, ri qua trung im, mi nh vy.Hay ni cch khc, ta khng c ln th c nh, ca ta s ln mi ln mi. Cc nh khoa hckhi y ni rng s tin v v tn (k hiu l ). V thc t chng minh rng khong cchn mt lc no s bng (mi tn trng ch).

Tm li, khi s ln (qua trung im) c ln mi, ln mi, n s t c mc ch ca n lkhong cch s bng . Ta gi y l gii hn ca khong cch khi ( c ln mi).Gii hn ny tn ti v thc t ch ra mi tn chc chn trng ch.

S d ta nhc ti cm t gii hn tn ti y v s c trng hp gii hn khng tn ti, chngta s tm hiu sau.

Khi y cc nh khoa hc k hiu cho ngn gn trong ton hc l

c ngha l gii hn ca (dy) khi l .

k tip theo, ta s tip tc tm hiu thm mt vi nghch l khc v xem xem mi lin h giacc nghch l y vi nh ngha ton hc hon chnh v gii hn m chng ta c hc l g.

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Dinh Anh Thi : Tc gi blog, xut thn l sinh vin s phm ton, hin ang l nghincu sinh v Ton ng dng ti Php. Chnh v nim am m Ton hc, tin hc ng dngv qua trng lp s phm nn Thi rt quan tm n vic ng dng Ton hc, Tin hc

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10.9.14 , Hiu ton hc Ton s cp

Vit bi cng blog bng cch lin h [email protected].


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Bi ng Mi hn

v qua trng lp s phm nn Thi rt quan tm n vic ng dng Ton hc, Tin hcvo trong cuc sng thc t cng nh vic truyn t kin thc Ton cho hc sinh theomt phung php hin i v tch cc nht, gy hng th v hiu su cho ngi hc,ngi c ni chung. Mi chi tit xin lin h [email protected]. Blog c nhnAnh Thi blog.



Tuyn tp cc eBooks hiu thmv th gii ton hc k th (cp

nht ngy 10/12/2014)

Tnh hp l ca nghim hphung trnh v khi nim

'condition number' ca mt ma trn (k1)

vui: Anh chng chy xe (pn)

Cc website hay dnh cho vichiu ngha thc tin ca ton

hc (update 12/9/2014)

Hiu v khi nim 'gii hn' trongton hc (k 2)

Gii thiu

Math2IT l chuyn trang v Ton hc+ Gio dc v Cng ngh. Mongmun chia s tri thc v kht khao imi nn gio dc nc nh. cthm.

Theo di chng ti

+38 xut url ny trn Google

Nhiu nht 7 ngy qua

Nhng cch v hnh chnh xcbng thc k v compa

Nhng vic cn lm sau khi ciUbuntu 14.04 v mt s phin

bn khc

Nhng phn mm nn ci choUbuntu, Elementary OS, Linux

Son tho ton hc trong LaTeX

Math2ITTon&Tin1,461Like



(k 1)

WolframAlpha : gii ton theotng bc

Tng hp nhng vic cn lmsau khi ci Windows c mt

chic my tnh y mnh m

Ti liu tham kho dnh cho itng t hc LaTeX (cp nht

ngy 6/11/14)

Bi ngu nhin

Tnh hp l ca nghim hphung trnh v khi nim'condition number' ca mt ma

trn (k 1)

Nhng vic cn lm sau khi ciElementary OS

Tips: Cch khc phc li khngnhn m thanh khi kt ni HDMItrong Ubuntu, Elementary OS

Nhng bc hnh ng gii thchcch vn hnh ca nhng vtquen thuc xung quanh bn

Tuyn tp nhng cng c trctuyn gip bn thit k mt bnCV (l lch) tuyt p

Nhiu nht 30 ngy qua

Nhng cch v hnh chnh xcbng thc k v compa

Nhng vic cn lm sau khi ciUbuntu 14.04 v mt s phin

bn khc

Nhng phn mm nn ci choUbuntu, Elementary OS, Linux

Ti liu tham kho dnh cho itng t hc LaTeX (cp nht

ngy 6/11/14)

Son tho ton hc trong LaTeX



Son tho ton hc trong LaTeX(k 1)

WolframAlpha : gii ton theotng bc

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Hiểu về khái niệm 'giới hạn' trong toán học (kỳ 1) _ Math2IT.pdf