0002000201000aLimit2.ACT000102000alimit2.EAC010000002905  limit2.EACLACT),0C~% 'A[4 'Finding es us geometric approach/\7 AuthorNoProfessor Wei-Chi Yang Radford University 9 e-mail: wy0@r).edu<!URL: http://www.#/5! Objectjs:[1. will see how we can f  ax-100 geometrically.[02. We will see how we can0neralize the result.[;ExampleH . Explore  #]hxp-10 ^ation:[`RStep 1drawfollowing two polar equ:s by us+y ߉ s+p: r=n6 and r=1 (< use r=n, tlot+in CP)nalso point 9 # into the geometry strip.[/Step 2. We pick a po= C on :curve C1: r=Un^'; and animate7along:]r+editZ4ihsokau approaches 10 from above. [NoteMw҈3 -number ofepdbe 49t0=0. 1=0?.]]J,% connect OChersf circle at B.4#a3a<u-vGor O+Key: 1. We note BC= x -1 and thegle -x= rc of BA. \EPlay3 imation=>Έ]ag pIu `r8f YP%@ Dv# r  A kL'C w xy q  `.A6AE 9 2C5U@ePG3rxD Angle of r:   !r@(6/YCpW: Y 3Qs  K[#`dOC. Bz C@ɐ Ƙo 91)U@e&pG3rx* Length BCso~ HC44I$!wS$ iq6g   %@#  ; @J6Q`&QUD Y 2c#"PWJ`J`z"p 1@ !^th(10p` FH OO YCpW3QsF%H\x\ 5Ƙ" ! 1 (1 gfE2 0 03333V g |  1=[UStep 4. We collect the f owing information after ani. [Note1 at we pick>\0numb0of sfs to be 49 and t0=0. 1 ] [/umn1= angle approacheB0 fromright, "8 2= distancezBC which is x-1[10.49 0.6322554848 4797916667& 1572541659 4695833333L 5993627526&59375206s4444ln 5670344031438o 551067379&28m35359515418Ƙ 51971523740٘04324880398128909038638HI 47391755763777q 4589910010.3675 0.4441248654 3572916667& 2950002429 3470833333L14934381l368n 006051556[32jmn 386421047n1645o 3722943469&062319694529604 344555792128k 330939302275n 174607[2UGH30403Z852552IZ0520.240.2776778243  347916667 64738621%245833338 518517786&14375Z91741192H04km2654181939o914126359n0.18m0175522G735om18Y489716ؘۆ463820153121662qH142Gۆ5663427I132701419720.12l 302731035 0.112291666788266721 020833333& 0742659250.09187 62804195I8lop( 851100498<7145r4P 7405351140.061r 6310965069J51r52405984r686157306rq781r2q 205662348V942709d00\4+Look at the scatter plot=>Column C against  A.  N SpreadsheetS 1 @ 1@@& +yyfpiX3306Y7PQIf 8Q(uT3 wpgDW)DGDn6Pn&ffuEu %PQ 3QuboeAuU E4y$X7DDDuDsTێ~~c3 SP46B2p "P) uQf*@6QAQQ%!330  @ 112%T11rTP6'R`6c$@QgDlQsy55qR72H (suWX)DHe)P$ _C `QU`*BG r)CF6X9iEPQDUWl0~F$fh0%@wgx$6ds!lQx9A)&TwacYzuR!6XHKKK wF8 #eFbU 6Sf4'6Ar`l0'1PfrBe' bA6Q6@SQ61 erQ$ QYlq^rQf#H' P  @ 1`z! '#^#i#M XF# i ( 0; 8^d  # B#Frrrrrrrr^# !"#$ %  `#&# #'F(i)*+,-8.@;/H^0P1X1 >(p `i% Tu8w% YH8d3:14@F @B[27Step 5. We conjecture from the above scatter plotat  BMxU-10+w=1.['Ti!following represents distance BC=g--1 when ngle x approaches to 0 but less :1than zero. **However, CP records positive angle. [:[ NoteI P t we pick e number of steps to be 49Pd t0=-0. 1=0 for7danimation above][| [1 0.3878900471112184474790266 0.368393913861:8215%55323192p48686O842023332835291&8929093L215786341463788707495596J0.3004547166 0.2932978542 860229055& 7871839499 713391563L63884429562132139 486509364 409163828L33150398r253049617381[22795_01O593r 192909488 18465235011762589167851Z17803V150662L41974299VA196063 1243280267 1153136606 0.1062065064 0.09700561119 8776730226( 7837659367 6894769281P#32575d 497398308P3995f89x30196656d 2021326913 101271746P0[Note that when x<0, x' -1<0 (so :'&-ِ is positive).\Kx goes to 0 but less5n 0=>see cell C48 for numerical value of (^x-1)/x剭N SpreadsheetS 1 @ 1@@& +yyfpiX3306Y7PQIf 8Q(uT3 wpgDW)DGDn6Pn&ffuEu %PQ 3QuboeAuU E4y$X7DDDuDsTێ~~c3 SP46B2p "P) uQf*@6QAQQ%!^330  @ 11G11Dtyf6h99Qa!PlU21EHhgGӢB32Q5)G(D !W4cxIUlEG`_)xT )P xqq3V06cD%QV!2qHe 6@@c3%0If`3 2'#5e#Pv%zg_Y1x`< Pf6BABP63`cQ$2&p?16` e@ pawg0"QvY66Gi(Qc%u l9lgWgef&0'F`  @ 1`z! '#^#i#M XF# i ( 0; 8^d  # B#Frrrrrrrr^# !"#$%`#&# #'F(i)*+,-8.@;/H^0P1X@R u x-10R91[CD Exercise 1. VAU"the Geometry Strip above to explor$following two limits:]a)ʭsin(x) andSb) TcosT [R [  Example 2.[ By using the substitution u=Cx/ -1 or u+1i&, we get x=ln(% ). We note_a->0 iiesu ThereforeQ expect uc0=1. Gemeraliza:GLr=f(n)u f is diffntiab0 an interval contain40d  if(x  n y B-1x0 =constant. [,WVRemark: The generalization above follows directly from L'Hopital's Rule, which we can ^ Xcompute t^Έ e `@Gs# .i! Dv r  Av6L'CC uxy qq@  u# B@6 F'`) 7uQC$ 49  BqW$h!6 Y (@9S"sBrsiqY3{H BC %9EIPyyQr@ CGG@ YDy"1q*ɔ@1(10qyXH"s#"v(03x&PGD)T$B `A+%@  $#&0 '"5@#"(fRsin(8th/5)+1` xP IXP9 qhcYĈ!ۈ %ʍ[%$column1=angle x (measured in radian),,2=length BC= f(n)-1[T 0.3490658504 5299194953& 306939635 0.504765544 0.3123220767 4791756966& 29395018989 4531718525 0.275578303^ 426776456 257206416140003 2388345292^ 372902579^4626424547_5 20209075577402218386889735Փ. 16534698126147999r 1469750943299873.12860320S 2043162121102313 0.1754571618 0.09185943432'464465('7348>745' 173093406N 5511566059b 880708108 3674377373(5a61912P 183718868d 29390820805.105o655\52Look at C19=1.59977149 which is close to the limit )6?N SpreadsheetS  @ @ IXP +0i9cP2 vp6QuWlW d8E)  F&B Uqh`e4iFP)(`2#!2 YCCz4TtP @QfgCw70f Yh`g)LvUDyVg6SRӠQ&wdVl##pr%yEGE@t$sRuaGQ22J1buEqaFDe6-0@`QplV6 6PPA`z!  ###M#XF# iCCb#2# F J J  rz;^ @re[$Method 2. Algebraically.R$define f(x)=sin(8x/5)+1R#done3 Y.-1fx0<"d1.6`R define g(x)=cos(8x/5)R!done1 3,-1@x0<0[I Exercise 3.[{Consider r=2*n! and . Find 9./-2 graphically.R 2[neActfH xL(8*x/5)+1gH xP(8*x/5)010008main.ACT0001020012eActivity Save.EAC010000002905  limit2.EACLACT),0C~% 'A[4 'Finding es us geometric approach/\7 AuthorNoProfessor Wei-Chi Yang Radford University 9 e-mail: wy0@r).edu<!URL: http://www.#/5! Objectjs:[1. will see how we can f  ax-100 geometrically.[02. We will see how we can0neralize the result.[;ExampleH . Explore  #]hxp-10 ^ation:[`RStep 1drawfollowing two polar equ:s by us+y ߉ s+p: r=n6 and r=1 (< use r=n, tlot+in CP)nalso point 9 # into the geometry strip.[/Step 2. We pick a po= C on :curve C1: r=Un^'; and animate7along:]r+editZ4ihsokau approaches 10 from above. [NoteMw҈3 -number ofepdbe 49t0=0. 1=0?.]]J,% connect OChersf circle at B.4#a3a<u-vGor O+Key: 1. We note BC= x -1 and thegle -x= rc of BA. \EPlay3 imation=>Έ]ag pIu `r8f YP%@ Dv# r  A kL'C w xy q  `.A6AE 9 2C5U@ePG3rxS Angle of r:   !r@(6/YCpW: Y 3Qs  K[#`dOC. Bz C@ɐ Ƙo 91)U@e&pG3rx* Length BCso~ HC44I$!wS$ iq6g   %@#  ; @J6Q`&QUD Y 2c#"PWJ`J`z"p 1@ !^th(10p` FH OO YCpW3QsF%H\x\ 5Ƙ" ! 1 (1 gfE2 0 03333V g |  1=[UStep 4. We collect the f owing information after ani. [Note1 at we pick>\0numb0of sfs to be 49 and t0=0. 1 ] [/umn1= angle approacheB0 fromright, "8 2= distancezBC which is x-1[10.49 0.6322554848 4797916667& 1572541659 4695833333L 5993627526&59375206s4444ln 5670344031438o 551067379&28m35359515418Ƙ 51971523740٘04324880398128909038638HI 47391755763777q 4589910010.3675 0.4441248654 3572916667& 2950002429 3470833333L14934381l368n 006051556[32jmn 386421047n1645o 3722943469&062319694529604 344555792128k 330939302275n 174607[2UGH30403Z852552IZ0520.240.2776778243  347916667 64738621%245833338 518517786&14375Z91741192H04km2654181939o914126359n0.18m0175522G735om18Y489716ؘۆ463820153121662qH142Gۆ5663427I132701419720.12l 302731035 0.112291666788266721 020833333& 0742659250.09187 62804195I8lop( 851100498<7145r4P 7405351140.061r 6310965069J51r52405984r686157306rq781r2q 205662348V942709d00\4+Look at the scatter plot=>Column C against  A.  N SpreadsheetS 1 @ 1@@& +yyfpiX3306Y7PQIf 8Q(uT3 wpgDW)DGDn6Pn&ffuEu %PQ 3QuboeAuU E4y$X7DDDuDsTێ~~c3 SP46B2p "P) uQf*@6QAQQ%!330  @ 112%T11rTP6'R`6c$@QgDlQsy55qR72H (suWX)DHe)P$ _C `QU`*BG r)CF6X9iEPQDUWl0~F$fh0%@wgx$6ds!lQx9A)&TwacYzuR!6XHKKK wF8 #eFbU 6Sf4'6Ar`l0'1PfrBe' bA6Q6@SQ61 erQ$ QYlq^rQf#H' P  @ 1`z! '#^#i#M XF# i ( 0; 8^d  # B#Frrrrrrrr^# !"#$ %  `#&# #'F(i)*+,-8.@;/H^0P1X1 >(p `i% Tu8w% YH8d3:14@F @B[27Step 5. We conjecture from the above scatter plotat  BMxU-10+w=1.['Ti!following represents distance BC=g--1 when ngle x approaches to 0 but less :1than zero. **However, CP records positive angle. [:[ NoteI P t we pick e number of steps to be 49Pd t0=-0. 1=0 for7danimation above][| [1 0.3878900471112184474790266 0.368393913861:8215%55323192p48686O842023332835291&8929093L215786341463788707495596J0.3004547166 0.2932978542 860229055& 7871839499 713391563L63884429562132139 486509364 409163828L33150398r253049617381[22795_01O593r 192909488 18465235011762589167851Z17803V150662L41974299VA196063 1243280267 1153136606 0.1062065064 0.09700561119 8776730226( 7837659367 6894769281P#32575d 497398308P3995f89x30196656d 2021326913 101271746P0[Note that when x<0, x' -1<0 (so :'&-ِ is positive).\Kx goes to 0 but less5n 0=>see cell C48 for numerical value of (^x-1)/x剭N SpreadsheetS 1 @ 1@@& +yyfpiX3306Y7PQIf 8Q(uT3 wpgDW)DGDn6Pn&ffuEu %PQ 3QuboeAuU E4y$X7DDDuDsTێ~~c3 SP46B2p "P) uQf*@6QAQQ%!^330  @ 11G11Dtyf6h99Qa!PlU21EHhgGӢB32Q5)G(D !W4cxIUlEG`_)xT )P xqq3V06cD%QV!2qHe 6@@c3%0If`3 2'#5e#Pv%zg_Y1x`< Pf6BABP63`cQ$2&p?16` e@ pawg0"QvY66Gi(Qc%u l9lgWgef&0'F`  @ 1`z! '#^#i#M XF# i ( 0; 8^d  # B#Frrrrrrrr^# !"#$%`#&# #'F(i)*+,-8.@;/H^0P1X@R u x-10R91[CD Exercise 1. VAU"the Geometry Strip above to explor$following two limits:]a)ʭsin(x) andSb) TcosT [R [  Example 2.[ By using the substitution u=Cx/ -1 or u+1i&, we get x=ln(% ). We note_a->0 iiesu ThereforeQ expect uc0=1. Gemeraliza:GLr=f(n)u f is diffntiab0 an interval contain40d  if(x  n y B-1x0 =constant. [,WVRemark: The generalization above follows directly from L'Hopital's Rule, which we can ^ Xcompute t^Έ e `@Gs# .i! Dv r  Av6L'CC uxy qq@  u# B@6 F'`) 7uQC$ 49  BqW$h!6 Y (@9S"sBrsiqY3{H BC %9EIPyyQr@ CGG@ YDy"1q*ɔ@1(10qyXH"s#"v(03x&PGD)T$B `A+%@  $#&0 '"5@#"(fRsin(8th/5)+1` xP IXP9 qhcYĈ!ۈ %ʍ[%$column1=angle x (measured in radian),,2=length BC= f(n)-1[T 0.3490658504 5299194953& 306939635 0.504765544 0.3123220767 4791756966& 29395018989 4531718525 0.275578303^ 426776456 257206416140003 2388345292^ 372902579^4626424547_5 20209075577402218386889735Փ. 16534698126147999r 1469750943299873.12860320S 2043162121102313 0.1754571618 0.09185943432'464465('7348>745' 173093406N 5511566059b 880708108 3674377373(5a61912P 183718868d 29390820805.105o655\52Look at C19=1.59977149 which is close to the limit )6?N SpreadsheetS  @ @ IXP +0i9cP2 vp6QuWlW d8E)  F&B Uqh`e4iFP)(`2#!2 YCCz4TtP @QfgCw70f Yh`g)LvUDyVg6SRӠQ&wdVl##pr%yEGE@t$sRuaGQ22J1buEqaFDe6-0@`QplV6 6PPA`z!  ###M#XF# iCCb#2# F J J  rz;^ @re[$Method 2. Algebraically.R$define f(x)=sin(8x/5)+1R#done3 Y.-1fx0<"d1.6`R define g(x)=cos(8x/5)R!done1 3,-1@x0<0[I Exercise 3.[{Consider r=2*n! and . Find 9./-2 graphically.R 2[neActfH xL(8*x/5)+1gH xP(8*x/5)030197002000080000000022cax{p\՝{nrnKm[~@dKlMkJdF~!-k–ͺVD[!;."q\Jyΐ@bw3PnUrfS#;},$ls}~-s^%U6J>c:|C!ऱ0>w踌T67x4y~9}u݉xn, tݞj!ET5vϐcG6u.N $ĺ| .IlCc%3G~Ym۽GS?_:te,U ւNr ^wvш67jsJ3hL~ԩg>=[-Rrc X$/\_lv[Hxm%7X;\k욊 [Wte-Nk=%MOV&)P_'"oߨe2ϵJќnҗKM#sk\>٭ڬ&l *4+O";&32w& 7r!>GJb6L|]yC7 :m]jX{dtoYK>R[ZQЖ,A\9|WRWͱr}n囬h-Xt2Ԟy-VꅼԻ7v^/1%ٖ^Rpszɛ 5L:pn6|C9=/ʳ~/dgD:w 5=ni4u,gJwMH{-wDy3&ZJOG`ܲ j2iqCiKPyK J Hi?BeǤO5.)Թo{\_<}acx_9f./Sz'+쯵З7w o3e͚Ng볯~y S;ҠG)A)R"R:eF׼O47J㱺xxnAաskܳZ\G.4;]?Ϸ瞴֪pTdEY׉P~ vL%]2ݞTB?d} ByP..MfOeV=x7f=_muB=p@YܘzD1oY 5Üw0fMԓfI KfYDh]81kh˺,.])-RޑHJQb)פ,+)KeM 2pIn6dMP[fSF" Md`y#mf2~l'7M2pN rz4&M2 EXy`6"Nh6} 6TZntVK&[kL)k\3)Q)+nK{24/YvXs$[SB]Ҡn, j)wHeNiP~ө[f9B]Ҡ l5OҠ2# j([Aͯ6iP2ԫ:5my|G\ISgg6ݍrX=Pi*;WϥUZ& c*d8>ƅ쬐ߗڥ3fHoJBkq m uNr?>V2 DZztn,w7Z49vxt[gȾшQm]w~Ozo~plՆʻ_/'ݥ̱f߾Uaַ*1_,j\P)kgim/~,hίo"6_{>{׭r+wv} ƿ091Ky'¥[RNFU?ߪRϷKewKN'(9:a_zsĹ >aS15a +DkNÉka3Ua˥ -\>j"Bӹ٧*#}ҼpnIy@oi-fO!00%7o'#MLɀKtNyw"p.RhmC8]_韞*L;ۥ<v~쿻˗*|ӳ_Yt5~ߞWj"/ީWeRSs-rOsiRg~Z~>3?I}4;әz?1#?Z~58#?]~U}w}dycO/Iz(VOFvZdžwy-MR'Ck*TVXk6-j^lꐎؤVS{?͕դbztO<_H$J";QIdJ"$2^7HRt))oKMwtSXom;yjN`ҩJ lj33؉J¸EwR n}m9O.;Oiv|V%}uB%KK:}MN.TBNJXIi|tΩ%fN+)}73TژBuBz;t*L)qht")gPVSHgmRtR.eҴ5f ֲV&+힮dsu^'R98!IK:O7tʒrtD9ɿSI?K$I nrˬ4ѩЗ.Y:L&V8'c;|Q߰[~y_޿;P5QITh޺tN95br)9#GuQo3?sW3v*wV-'XJ .V^۩_,Y֜ZF,4+o.$?\sw$ 5l 1brZ>q[Z_Y9֎*: ==dpV8kշ/OWWԎB@-jIam #I;d ӑ3vj|W5b2Ɍ\@ |g>޽v+7չ#>{6k{[Oζ^Z^4e#$8SN-~^零;m[x蠾]e]ZawW.$,%mJeu{rnƻzj>pa={o޽rZDذqkur}߱{cCTܷf_ڀc-׻P/agDkyfDgr\Quףsˑ i-uTFKcXֳZ*0_Ӫ[%\c >_.?V<|`ЁlP[Jj2r`-5;CTKU/IߏIEj QCC9YϤT2H:Bh,I$P:NFTJeD,8BL6S, %T<I&JdRD6&JDH*XR<dbш 1Q6jŢt&O&&D&OBt2KEYGH*I3cdx6ryi4-9+S{+߆zJ6+7PE U:wჇZ\~`뮾.ۼC|d+3.XPO=.w*OfHeeltOzѣG8&4S5ogԙI~I"o|8888w?αϱzui#|gy8~Cjm5Ozwo 18'5ʴV)r"<>ƺ?yt /lm~m8MS~?gg.X6={H>) 弔W\򶔫Re+_JN)=Rz I9)~vUM~ái4>Pmzpq>C7jw~1sعu⿫iߑw_dž96jŝ˯~#}D>"G|}Do|/z6Ρc㪺#Ob5!n\ڙYϧ 93tYF#$]kwguG$ذVM/D{z..4Z}/}нcϻݏܳ!w%N=x>qo_زm?q|ދr6ta:Tkލ^5ێ\~ً[=fHl%ڕ*nJA:ᾓ7mXrʳ0LgކC7{<#.jK_"֋)O'푷ͽH^SKe,,BHCmBuWyarOf4<1կًh0*knS 9ڔInyI*K ^f^Sz?j~J ^?^PMU~z/p[BvE(\iCx?*E]Vb.vq#حf70yϳNu!t"Ȥӵ#8$ Q)i}4؞dDZc6dI*.?ɴs{fd4t4mlfKD,hƱ=eW'K}sVKGqtIn8Y2)TTz/"MN&lF"hĹHT3&#)*dcJXKx\*t6Dc'I>ff\?_pVL-Hmmkw vFG @ъ;ڵ2"raY:v]>s; gGw=sO}ǟ9>Lԙʥ/@_^qrim(<(ߋzU#߲|8gGjv(!9BwOQGvNH1QDrH,:i VI)g(&sau#%u&ZXťOsL'AGIg#9giKBt'Ir!ȁq!H{RRZbiuwT4Rq9#|M'i,%r4%?9{qt .tγ*&;P#K%2R7;;RoIg&LƜg)rde v ĝz J&JDgfTd}Zsxܾ|:Ver~+*8ppN' v0U\Wq]uU\WqT_yQ0417