a lim x c f x L if and only if to each 0 there is a \u03b4 such that f x f c

A lim x c f x l if and only if to each 0 there is a

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(a) limxcf(x) =Lif and only if to each>0,there is aδ >0such thatD.(b) limxcf(x) =Lif and only if for each deleted neighborhoodUofcthere isa neighborhoodVofLsuch thatf(UD)V.(c) limxcf(x) =Lif and only if for every sequence{sn}inDthat converges|f(x)-f(c)|<whenever|x-c|< δ,xtoc,sn=cfor alln,the sequence{f(sn)}converges toL.(d) limxcf(x) =Lif and only if limh0f(c+h) =L.(e) Iffdoes not have a limit atc,then there exists a sequence{sn}inDsn=cfor alln,such thatsnc, but{f(sn)}diverges.(f) For any polynomialPand any real numberc,limxcP(x) =P(c).(g) For any polynomialsPandQ,and any real numberc,limxcP(x)Q(x)=P(c)Q(c).4. Find aδ >0such that 0<|x-3|< δimplies|x2-5x+ 6|<14.5. Find aδ >0such that 0<|x-2|< δimplies|x2+ 2x-8|<110.6. Prove thatlimx1(4x+ 3) = 7.7. Prove thatlimx3(x2-2x+ 3) = 6.8. Determine whether or not the following limits exist:(a) limx0sin1x.(b) limx0xsin1x.9. Letf:DRand letcbe an accumulation point ofD. Suppose that limxcf(x) =LandL >0. Prove that there is a numberδ >0such thatf(x)>0for allxDwith 0<|x-c|< δ.10.(a) Suppose that limxcf(x) = 0 and limxc[f(x)g(x)] = 1. Prove that limxcg(x)does not exist.27
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(b) Suppose thatlimxcf(x) =L= 0 and limxc[f(x)g(x)] = 1. Does limxcg(x)exist, and if so, what is it?III.2CONTINUOUS FUNCTIONSDefinition 3.Letf:DRand letcD. Thenfis continuous atcif to each>0there is aδ >0such that|f(x)-f(c)|<whenever|x-c|< δ,xD.LetSD. Thenfis continuous onSif it is continuous at each pointcS.fiscontinuous iffis continuous onD.THEOREM 6. Characterizations of ContinuityLetf:DRand letcD.The following are equivalent:1.fis continuous atc.2. If{xn}is a sequence inDsuch thatxnc,thenf(xn)f(c).3. To each neighborhoodVoff(c),there is a neighborhoodUofcsuch thatf(UD)V.Proof:See Theorem 1.CorollaryIfcis an accumulation point ofD,then each of the above is equivalent tolimxcf(x) =f(c).THEOREM 7.Letf:DRand letcD. Thenfis discontinuous atcif andonly if there is a sequence{xn}inDsuch thatxncbut{f(xn)}does not convergetof(c).Continuity of Combinations of FunctionsTHEOREM 8. Arithmetic:Letf, g:DRand letcD. Iffandgarecontinuous atc,then1.f+gis continuous atc.2.f-gis continuous atc.3.fgis continuous atc;kfis continuous atcfor any constantk.28
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4.f/gis continuous atcprovidedg(c) = 0.THEOREM 9. Composition:Letf:DRandg:ERbe functions such thatf(D)E. Iffis continuous atcDandgis continuous atf(c)E,then thecomposition ofgwithf,gf:DR,is continuous atc.Proof:Let>0. Sincegis continuous atf(c)Ethere is a positive numberδ1such that|g(f(x))-g(f(c))|<whenever|f(x)-f(c)|< δ1, f(x)E. Sincefiscontinuous atcthere is a positive numberδsuch that|f(x)-f(c)|< δ1whenever|x-c|< δ, xD. It now follows that|g(f(x))-g(f(c))|<whenever|x-c|< δ, xDandgf
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