PDF-II.Evaluatethelimit,ifitexists1.limx!1x24x

Author : phoebe-click | Published Date : 2016-03-22

x23x42limx0p 2xp 2x x3limx93p x 9xx24limh0xh3x3 h5limh0xh2x2 h2 Answerlimx31 31 x 3xlimx3x3 3x 3xlimx3x3 3x1 3xlimx31 3x1 331 96limh0p xhp x hAnswe

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II.Evaluatethelimit,ifitexists1.limx!1x24x: Transcript

x23x42limx0p 2xp 2x x3limx93p x 9xx24limh0xh3x3 h5limh0xh2x2 h2 Answerlimx31 31 x 3xlimx3x3 3x 3xlimx3x3 3x1 3xlimx31 3x1 331 96limh0p xhp x hAnswe. 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 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 Defn:limx!af(x)=Lifforall0,thereexistsa0suchthat0jxajimpliesjf(x)LjShowlimx!12= Defn:limx!af(x)=Lifxclosetoaandxaimpliesf(x)isclosetoL.Defn:limx!a+f(x)=Lifxclosetoaandx moreexamplesoflimits {TypesetbyFoilTEX{1 SubstitutionTheoremfor TrigonometricFunctions lawsforevaluatinglimits {TypesetbyFoilTEX{2 TheoremA. Foreachpoint c infunction'sdomain: limx!csinx=sinc;limx!cco 1 2FIRSTTHINGSFIRST(5)Duringclasstheinstructorhasthenaldecisionondeterminingwhetheranar-gumentmaystandornot.Hisverdictmaystillbechallengedafteraproofis\published"(seerule(6)).(6)Ifsomeoneothertha -sizedPDF12lesoNorthAmericansshouldtakecarenottoinadvertentlygenerateletterpaper-sizedPDF12lesThispapertemplateshouldpreventthatfromhappeningifthepdflatexprogramisusedtogeneratethePDF12leTheabstractsh CONTENTSvChapter16.APPLICATIONSOFTHEINTEGRAL12116.1.Background12116.2.Exercises12216.3.Problems12716.4.AnswerstoOdd-NumberedExercises130Part5.SEQUENCESANDSERIES131Chapter17.APPROXIMATIONBYPOLYNOMIALS1 Introductiony=fxLimits&ContinuityRatesofchangeandtangentstocurves2 Averagerateofchangey x=fx2fx1 x2x1(1)Equationforthesecantline:Y=y xxx1+fx1Tangent -secantinthelimitx2!x1(Or,x!0).Slopeoftange (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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