00020002010010EpsilonDelta.ACT0001020017EpsilonDelta_Proofs.EAC0100000034a0 EpsilonDelta_Proofs.EAC ForKarla.ACT+. ;?R`E 'm[lAuthorDiane Whitfield%dw @casio.coAd[J5How to do and underst k-j (e-d) p،E NP What isN IN definition? I* very formal  used3e the existence of a %limitB f(x) a specific x-value.!/ The definition is often used to 6proof t-exis! ce or non-f a giv?limit.[>}Before looking atRk-jHB, let me review and stress a mportant pertyfunc, say, y=f(x).\"PleasR ad ---> tapic툷@%OFor y=f(x), x is the indepent variable. Thmean% at we have2freedom WHto choose anyDlue0waRfsx. Ifx 0 bounce ard in:interval (1,39can dobecaulit–.  PWamtless tunate y? yhighly onand lawlmustfollows already been set. ~{{example,t =4x8.  value as x->4?, means that x could be I4+.00001 or 4- ...1e choose of x is up to us. But,g we bounce QUaround>4 without touching i=Oy forcedU approch a s"l=? If itPis,enwsayԈlimitf(x)thR.R, on Pother hand,does nota`xSscl-r#]P4 from any directionyns somehow gaina level of indepe#nce if only Sfor the x-value04. Whenis hap<s, y may be heading to + @-jumpZ> a new spot and we sDthatt limit does n#exis)t x=4.[[ *Remember, youvefreedomchoose xlany becau!ts"t nature ith practicej will leardinter<s about-T=cp?simplife_oblem!L3**Also remember, the k-j definition is used to ei"r prove or dis a  given limitJ. ItEalJin many formalOofs and a general understanding of Q)bencial veryone.[Dێ pLet fEfuncan opinterval contac (except possibly at c)lXLXreal nug. 54Txtnt  :f(x)ExMcU#=L means that ["for each k>0ere exists a j*[4su0Fif 0<Kx-c" ok  c$6Lb}ͨ0P}@Wv}}I ^ +#4Ee`<}}"3kי)A_}! }yVq^YrV!}} <,K$6Oh};wVw} 1E<`{}cw1 1F[a{<}ӆCx; j0.d}?}}w6#4Eq\[ }֘ JS}2 Ш+<;Mau Ȩ}}p2 ̩}21̍B0 Ϙ͐/}}}3@ĿAe`8 w~$PD 6Ut;}}Hp}}m@}@wv@w @3VD! i|ľ~H|} }P̌G φΔj&1dl}}}x<в5Rk}[q\5G{}}!w n  6U}+}f@ `aw"} }  "?%"y8y< {Q? n}@",A  *" u} U""(A}})"}Ŋ}} "(E@"  @"( "7  %w!89p@:+g}^?}A}V @%!!E2Wu}}?w6   AG}Ke6 @ %9"2B}P } }}!eqǀS}w!I"($}rm  !I"$Qs}IA}}H }\Ԕq׎@}ϊwm 4$);M\r}ä} w6 &!J\}߷\ϔ/ #4E7Yj{}}Cw1\'`$6U}0OwBx} }}- .E\w}}+ Oa}䭇!aw 7#4E\[l}} .?w2%!8BJ7}J[lww7 1;G}}? }}- !3O|}}ࣘ?0dTf};҈uÇ`Ow  !1AZ }0c# /0*##Oe{$1̬}Mcywwc  Gkl}}}fww ! %6T }}I `H 8wH#5Yk} @  <@}}C}}w^'@1st Rewrite 0<fx-c form aherPequality containing#Xx-c4th7 Sjs C& ;. *Be careful with flippies. For a1/bC5th8 Set your results from step 4 equal to k. Solve for j.[@[J]*Remember, x is the indepent one. We needSf%distance"5x-ax@(j){make f(x)-L90. How small must j beguaranteOO?! Example 1[ Prothat  xP2=4 3\1st 3BWe must find a value for j in terms of k so that [x 2-4<3 whenever 0<3x-+0 dED1Ҏ<x-4Y0... Ok, let us choose k=.01, then j /5[i*For any&@try=anging7 value ofL to .0P.00Dd press EXE after eachIe.Rouk:R"0 / j:=k/50232a solve(x-20 d j=6k, %1x-38j sinceQ?.0ˈ=AR k:=.0001R"14!+ 3 j:=1/k21C01de solve(tx-3~ tapic 툿@%OFor y=f(x), x is the indepent variable. Thmean% at we have2freedom WHto choose anyDlue0waRfsx. Ifx 0 bounce ard in:interval (1,39can dobecaulit–.  PWamtless tunate y? yhighly onand lawlmustfollows already been set. ~{{example,t =4x8.  value as x->4?, means that x could be I4+.00001 or 4- ...1e choose of x is up to us. But,g we bounce QUaround>4 without touching i=Oy forcedU approch a s"l=? If itPis,enwsayԈlimitf(x)thR.R, on Pother hand,does nota`xSscl-r#]P4 from any directionyns somehow gaina level of indepe#nce if only Sfor the x-value04. Whenis hap<s, y may be heading to + @-jumpZ> a new spot and we sDthatt limit does n#exis)t x=4.[[ *Remember, youvefreedomchoose x any becau!ts T*t nature ith practicer will lear$linterDs aboux=c%^?simplife_oblem!7L**Also remember, the k-j definition is used to ei"r prove or dis a S given limitN. ItMatRin many formalWofs and a general understanding ofU%bencial veryone.[D[TLet fFfunc an opinterval conta(c (except possibly at c)l_L_real nuw. Ttnt  Bf(x)Mxc=L means that ["for each k>0ere exists a j*[Jsu0Fif 0<ax-c" ok  c$6Lb}ͨ0P}@Wv}}I ^ +#4Ee`<}}"3kי)A_}! }yVq^YrV!}} <,K$6Oh};wVw} 1E<`{}cw1 1F[a{<}ӆCx; j0.d}?}}w6#4Eq\[ }֘ JS}2 Ш+<;Mau Ȩ}}p2 ̩}21̍B0 Ϙ͐/}}}3@ĿAe`8 w~$PD 6Ut;}}Hp}}m@}@wv@w @3VD! i|ľ~H|} }P̌G φΔj&1dl}}}x<в5Rk}[q\5G{}}!w n  6U}+}f@ `aw"} }  "?%"y8y< {Q? n}@",A  *" u} U""(A}})"}Ŋ}} "(E@"  @"( "7  %w!89p@:+g}^?}A}V @%!!E2Wu}}?w6   AG}Ke6 @ %9"2B}P } }}!eqǀS}w!I"($}rm  !I"$Qs}IA}}H }\Ԕq׎@}ϊwm 4$);M\r}ä} w6 &!J\}߷\ϔ/ #4E7Yj{}}Cw1\'`$6U}0OwBx} }}- .E\w}}+ Oa}䭇!aw 7#4E\[l}} .?w2%!8BJ7}J[lww7 1;G}}? }}- !3O|}}ࣘ?0dTf};҈uÇ`Ow  !1AZ }0c# /0*##Oe{$1̬}Mcywwc  Gkl}}}fww ! %6T }}I `H 8wH#5Yk} @  <@}}C}}w^'@1st Rewrite 0<fx-c 1/b 5th ;8 Set your results from step 4 equal to k. Solve for j.[@[SRP*Remember, x is the indepent one. We needRf%distance"5x-ax@Y (j)make f(x)-L90. How small must j beguaranteOO?2 Example 1 Prothat  xX2=4 ;1st 3BWe must find a value for j in terms of k so that [x 2-4<3 whenever 0<3x-+0 dED1Ҏ<x-4Y0... 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