00020002010011Epsilon_Delta.ACT0001020011Epsilon_Delta.EAC01000000294b Epsilon_Delta.EACACT*-7:>Q `E 'S[4About k-j Definition\& Author=^~~Professor Wei-Chi Yang Radford University 9 e-mail: wy9@r2.eduE!URL: http://www.#/5܎ Objectv:`0 ere existsj{su0Fif 0<qx-c& 0, we need to find jsuch that if 0<x-c $#aߞج` e5y  $Y,[[ Remark: %PGiven k>0, by solving g(x,k)<0 (if'able), we actually get to know where x gqshould be such that  f(x)xc =L holds. Exercise 1. If D=x2 5+1, and g=0.0fithe required jl0<x-2 Given M>07a jsuch at if a-xM RIf intrdocfuncg(x,M)=3-M1a(lookr xm)>0. We e2use the following examplo plor!is concept.[ E+2.  If f1(x)=x x-3&2/ and given M=10!3O &, how clo should x b3 from| #left (lessan 3) soat>M.[ Solution:/Step 1. We definenj,M) as's:Rnj 1R7doneHGn=-M2[jT#Step 2. We take a look of the graph y=f1(x) in#editor below and notice "that1deed when x->3-, J->.[+VIn opr words, we would li to prove givGa posit  numb6M2can fij>0 so* f 3-xM g,M)>0.\TЈN(FinaForm$NGraph2D& 33 LISTSYS@4< Modify A$<STATCALC hd< p\x Sequence,xSheetO|olveEqbwr(UptupFLG1 (<Lis{4DPicxViewWind_osveb;xy^ H6ܐ䊗@萴P   -Z  $02<HT`lx !"# ̆ %؆&'(L!)*  +l, T- 8.D0P1\2h3t45EFHIJȆKԆLMNO6Q RST@]I LU ^_F` Xab2 dp ͆|ΆВZ(׆ zنڒ=Ēےn| FinancialFormat    system]^_` a bk4^ @@0` `à8 @x !$ a MatDatab.EAC   X <,<^] (",W%F!  %T`UϪج`( e5y  ]Wad[W-+Step 4. We set j=3-2.945725462=0.054274538R77 )3-2.945725462R" 0.054274538BtS [& Conclusion:432 Given M=10^3, and if we choose x so that 3-x3-j),(KY*(f(x)>M. This can be verified as follows:[@Can1. I~pick x=X, f1L Gnot quite larger n 1,000.R|1#)$999. 9955 T8=2 a bits>131000. 033'8[ Exercise 2. % If f1(x)=x x-3&2/ and given M=10!5O &, how clofshould x be to 3 from the |#left (less than 3) so t>M.eActfH xx^(2)+1f1H$x((x)/((x-3)^(2)))gH$x,y (f(x)-5)-yg1H x,Mf1(x)-My010008main.ACT0001020012eActivity Save.EAC01000000294b Epsilon_Delta.EACACT*-7:>Q `E 'S[4About k-j Definition\& Author=^~~Professor Wei-Chi Yang Radford University 9 e-mail: wy9@r2.eduE!URL: http://www.#/5܎ Objectv:`0 ere existsj{su0Fif 0<qx-c& 0, we need to find jsuch that if 0<x-c $#aߞج` e5y  $Y,[[ Remark: %PGiven k>0, by solving g(x,k)<0 (if'able), we actually get to know where x gqshould be such that  f(x)xc =L holds. Exercise 1. If D=x2 5+1, and g=0.0fithe required jl0<x-2 Given M>07a jsuch at if a-xM RIf intrdocfuncg(x,M)=3-M1a(lookr xm)>0. We e2use the following examplo plor!is concept.[ E+2.  If f1(x)=x x-3&2/ and given M=10!3O &, how clo should x b3 from| #left (lessan 3) soat>M.[ Solution:/Step 1. We definenj,M) as's:Rnj 1R7doneHGn=-M2[jT#Step 2. We take a look of the graph y=f1(x) in#editor below and notice "that1deed when x->3-, J->.[+VIn opr words, we would li to prove givGa posit  numb6M2can fij>0 so* f 3-xM g,M)>0.\TЈN(FinaForm$NGraph2D& 33 LISTSYS@4< Modify A$<STATCALC hd< p\x Sequence,xSheetO|olveEqbwr(UptupFLG1 (<Lis{4DPicxViewWind_osveb;xy^ H6ܐ䊗@萴P   -Z  $02<HT`lx !"# ̆ %؆&'(L!)*  +l, T- 8.D0P1\2h3t45EFHIJȆKԆLMNO6Q RST@]I LU ^_F` Xab2 dp ͆|ΆВZ(׆ zنڒ=Ēےn| FinancialFormat    system]^_` a bk4^ @@0` `à8 @x !$ a MatDatab.EAC   X <,<^] (",W%F!  %T`UϪج`( e5y  ]Wad[W-+Step 4. We set j=3-2.945725462=0.054274538R77 )3-2.945725462R" 0.054274538BtS [& Conclusion:432 Given M=10^3, and if we choose x so that 3-x3-j),(KY*(f(x)>M. This can be verified as follows:[@Can1. 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