PDF-Defn:limx!af(x)=Lifxclosetoa(exceptpossiblyata)impliesf(x)isclosetoL.

Author : kittie-lecroy | Published Date : 2016-07-11

DefnlimxafxLifforall0thereexistsa0suchthat0jxajimpliesjfxLjShowlimx12 DefnlimxafxLifxclosetoaandxaimpliesfxisclosetoLDefnlimxafxLifxclosetoaandx

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Defn:limx!af(x)=Lifxclosetoa(exceptpossiblyata)impliesf(x)isclosetoL.: Transcript

DefnlimxafxLifforall0thereexistsa0suchthat0jxajimpliesjfxLjShowlimx12 DefnlimxafxLifxclosetoaandxaimpliesfxisclosetoLDefnlimxafxLifxclosetoaandx. Example1.Forthefunctionfgraphedbelow,ndthefollowing: - 6 ?yx01234567123456712345671234567 eCCCCCCseBBBBes@@@es1.limx!3f(x)=2.limx!3+f(x)=3.limx!3f(x)=4.f(3)=5 p x+53=limx!4(2x8) (p x+53)(p x+5+3) (p x+5+3)=limx!4(2x8)(p x+5+3) (p x+5)2+3p x+53p x+59=limx!4(2x8)(p x+5+3) x4=limx!42(x4)(p x+5+3) x4=limx!42(p x+5+3)=2(p 4+5+3)=12:Trythefollowingexerc DefnԆv &eOvTnusvreo༐ᄇpnᄑeᄑinOviᄇ nOvnԇlnᄍvbଇiOii̎riOgv &eԂvnOv̄uԍvr&evruԊnvesvñuԅei,voO 2+1 31 4+1 5toobtainanyrealnumberasthesum!Asforthesecondargument,onemightobject(asCalletdid)thatformnwehave1xm 1xn=1xm+xnxm+n+x2n;sothatbyl'hopital'srule11+11+=limx!11xm 1xn=m n:B x23x42.limx!0p 2+xp 2x x3.limx!93p x 9xx24.limh!0(x+h)3x3 h5.limh!0(x+h)2x2 h2 Answer:limx!31 3+1 x 3+x=limx!3x+3 3x 3+x=limx!3x+3 3x1 3+x=limx!31 3x=1 3(3)=1 96.limh!0p x+hp x hAnswe moreexamplesoflimits {TypesetbyFoilTEX{1 SubstitutionTheoremfor TrigonometricFunctions lawsforevaluatinglimits {TypesetbyFoilTEX{2 TheoremA. Foreachpoint c infunction'sdomain: limx!csinx=sinc;limx!cco Defn:IfXisstationarytheautocovariancefunctionofXisCX(h)=Cov(X0;Xh).Defn:IfXandYarejointlystationarythenthecross-covariancefunctionisCXY(h)=Cov(X0;Yh).NoticethatCX(h)=CX(h)andCXY(h)=CYX(h)forallhands Selected Exercises. Goals. . Introduce . big-O . & big. -. Omega. S. how . how . to estimate . the size of functions using this notation.. Copyright © Peter . Cappello. 2. Preface. You may use . Defn. : . Polynomial function. In the form of: . .. . The coefficients are real numbers.. The exponents are non-negative integers.. The domain of the function is the set of all real numbers.. -sizedPDF12lesoNorthAmericansshouldtakecarenottoinadvertentlygenerateletterpaper-sizedPDF12lesThispapertemplateshouldpreventthatfromhappeningifthepdflatexprogramisusedtogeneratethePDF12leTheabstractsh CONTENTSvChapter16.APPLICATIONSOFTHEINTEGRAL12116.1.Background12116.2.Exercises12216.3.Problems12716.4.AnswerstoOdd-NumberedExercises130Part5.SEQUENCESANDSERIES131Chapter17.APPROXIMATIONBYPOLYNOMIALS1 (x1)21A-3Identifyeachofthefollowingaseven,odd,orneithera)x3+3x 1x4b)sin2xc)tanx 1+x2d)(1+x)4e)J0(x2),whereJ0(x)isafunctionyouneverheardof1A-4a)Showthateverypolynomialisthesumofanevenandanoddfunction t esint�1dt.(b) 2R0ln(sint)dt(c)1R01 t2+p tdt(d)1R0cos1 t2dt.(e)1R0sint3dt(f)1R1sin2t p tesintdt(g)1R1tsint4dt(h) 4R0dt t�sint.(i)1R11�5sin2t t2+p tdt(j)1R0et 2 p 1�costdt(k)1R1t g(x)=limx!af(x) g(x)limx!ah(x) g(x)=00=0:Thisshowsthatf(x)h(x)=o(g(x)).Therefore,afunctionoftheformo(g(x))o(g(x))isalsooftheformo(g(x))andwewritethisfactaso(g(x))o(g(x))=o(g(x)).2.

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