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NATIONAL UNIVERSITY OF SINGAPORESEMESTER 1, 2020/2021MA1102R CalculusSolution to Tutorial 2REVIEWRecall theintuitive definitionoflimit: We write limx→∞f(x)=L, orxaf(x)L(xnegationslash=a)if by takingxsufficiently close toa(but not equal toa), the value off(x) is arbitrarily close toL.Mathematically, two numbersαandβare close if their distance|αβ|is small. So we havethe following observation:0<|xa|<δ⇒|f(x)L|<ǫ.We shall still consider the meaning of “f(x) is arbitrarily close toL”. This means that|f(x)L|can be as small as possible, in the sense that it can be smaller thananygiven positive numberwhenever|xa|(negationslash=0) is small. Therefore, theprecise definitionis:For anyǫ>0, there exists a numberδ>0 such that0<|xa|<δ⇒|f(x)L|<ǫ.In the proof using the precise definition, the key point is to find a properδfor the givenǫ. Weshall note the following:(i) The choice ofδis an expression in terms ofǫ, but not in terms ofx. For example,δ=ǫ+x2must be wrong.(ii)δrepresents the distance, so it must be always>0. For example, we cannot takeδ=ǫ1.(iii) Unlessy=f(x) is a constant function, whenǫ0 we must haveδ0 as well (cf. Q2). Forexample, one cannot takeδ=ǫ+1.(iv) The choice ofδis not unique. For example, in Q3(b), one may useδ=min{1,ǫ/7} orδ=min{2,ǫ/9}, etc..1
MA1102R CALCULUS TUTORIAL SOLUTION 22For the infinite limit limxaf(x)=∞, orxaf(x)→∞,(xnegationslash=a)it means that by takingxsufficiently close toa, the value off(x) is arbitrarily large. We haveunderstood the meaning of “xsufficiently close toa”. Then what is “f(x) is arbitrarily large”?This is simply to sayf(x) is very large, in the sense thatf(x) can be larger than any givennumberMwhenever|xa|(xnegationslash=a) is small. So theprecise definition of infinite limitis:For any numberM, there is a numberδ>0 such that0<|xa|<δf(x)>M.In the textbook, it only considers whenM>0; but this is not essential: If the statement holdsfor allMR, then in particular it holds for allMR+. Conversely, if the statement holds forallMR+, i.e., we can choosexclose enough toaso thatf(x)>M, then for anyM0,f(x)>M>M; so the statement holds for nonpositive numbers as well. Our definition justfollows the one from the textbook.From the precise definitions, we see that although the notations for limxaf(x)=L(whereLisa number) and limxaf(x)=∞are similar, they are different by considering the behavior off(x)whenxis neara. In particular, aninfinite limitis not a (finite)limit.

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