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Indeed, if limEx→a f (x) = A, then for any neighborhood V (A) of A there exists a deleted neighborhood Ů E (a) of the point a in Esuch that for x ∈ Ů E (a) we have f (x) ∈ V (A). If the sequence {xn } of points inE\a converges to a, there exists an index N such that xn ∈ Ů E (a) for n > N , andthen f (xn ) ∈ V (A). By definition of the limit of a sequence, we then conclude thatlimn→∞ f (xn ) = A.We now prove the converse. If A is not the limit of f (x) as E x → a, thenthere exists a neighborhood V (A) such that for any n ∈ N, there is a point xn in thedeleted n1 -neighborhood of a in E such that f (xn ) ∈/ V (A).
But this means that thesequence {f (xn )} does not converge to A, even though {xn } converges to a.3.2.2 Properties of the Limit of a FunctionWe now establish a number of properties of the limit of a function that are constantlybeing used. Many of them are analogous to the properties of the limit of a sequencethat we have already established, and for that reason are essentially already knownto us.
Moreover, by Proposition 1 just proved, many properties of the limit of afunction follow obviously and immediately from the corresponding properties ofthe limit of a sequence: the uniqueness of the limit, the arithmetic properties of thelimit, and passage to the limit in inequalities. Nevertheless, we shall carry out all theproofs again. As will be seen, there is some value in doing so.9 This proposition is sometimes called the statement of the equivalence of the Cauchy definition ofa limit (in terms of neighborhoods) and the Heine definition (in terms of sequences).E. Heine (1821–1881) – German mathematician.3.2 The Limit of a Function111We call the reader’s attention to the fact that, in order to establish the propertiesof the limit of a function, we need only two properties of deleted neighborhoods ofa limit point of a set:B1 ) Ů E (a) = ∅, that is, the deleted neighborhood of the point in E is nonempty;B2 ) ∀Ů E (a) ∀Ů E (a) ∃Ů E (a) (Ů E (a) ⊂ Ů E (a) ∩ Ů E (a)), that is, the intersectionof any pair of deleted neighborhoods contains a deleted neighborhood.
This observation leads us to a general concept of a limit of a function and the possibility ofusing the theory of limits in the future not only for functions defined on sets ofnumbers. To keep the discussion from becoming a mere repetition of what was saidin Sect. 3.1, we shall employ some useful new devices and concepts that were notproved in that section.a. General Properties of the Limit of a FunctionWe begin with some definitions.Definition 4 As before, a function f : E → R assuming only one value is calledconstant. A function f : E → R is called ultimately constant as E x → a if it isconstant in some deleted neighborhood Ů E (a), where a is a limit point of E.Definition 5 A function f : E → R is bounded, bounded above, or bounded belowrespectively if there is a number C ∈ R such that |f (x)| < C, f (x) < C, or C <f (x) for all x ∈ E.If one of these three relations holds only in some deleted neighborhood Ů E (a),the function is said to be ultimately bounded, ultimately bounded above, or ultimately bounded below as E x → a respectively.Example 7 The function f (x) = sin x1 + x cos x1 defined by this formula for x = 0is not bounded on its domain of definition, but it is ultimately bounded as x → 0.Example 8 The same is true for the function f (x) = x on R.Theorem 1 a) (f : E → R is ultimately the constant A as E x → a) ⇒(limEx→a f (x) = A).b) (∃ limEx→a f (x)) ⇒ (f : E → R is ultimately bounded as E x → a).c) (limEx→a f (x) = A1 ) ∧ (limEx→a f (x) = A2 ) ⇒ (A1 = A2 ).Proof The assertion a) that an ultimately constant function has a limit, and assertion b) that a function having a limit is ultimately bounded, follow immediatelyfrom the corresponding definitions.
We now turn to the proof of the uniqueness ofthe limit.1123LimitsSuppose A1 = A2 . Choose neighborhoods V (A1 ) and V (A2 ) having no pointsin common, that is, V (A1 ) ∩ V (A2 ) = ∅. By definition of a limit, we have lim f (x) = A1 ⇒ ∃Ů E (a) f Ů E (a) ⊂ V (A1 ) ,Ex→a lim f (x) = A2 ⇒ ∃Ů E (a) f Ů E (a) ⊂ V (A2 ) .Ex→aWe now take a deleted neighborhood Ů E (a) of a (which is a limit point of E)such that Ů E (a) ⊂ Ů E (a)∩ Ů E (a). (For example, we could take Ů E (a) = Ů E (a)∩Ů E (a), since this intersection is also a deleted neighborhood.)Since Ů E (a) = ∅, we take x ∈ Ů E (a). We then have f (x) ∈ V (A1 ) ∩ V (A2 ),which is impossible since the neighborhoods V (A1 ) and V (A2 ) have no points incommon.b.
Passage to the Limit and Arithmetic OperationsDefinition 6 If two numerical-valued functions f : E → R and g : E → R have acommon domain of definition E, their sum, product, and quotient are respectivelythe functions defined on the same set by the following formulas:(f + g)(x) := f (x) + g(x),(f · g)(x) := f (x) · g(x), ff (x)(x) :=, if g(x) = 0 for x ∈ E.gg(x)Theorem 2 Let f : E → R and g : E → R be two functions with a common domainof definition.If limEx→a f (x) = A and limEx→a g(x) = B, thena) limEx→a (f + g)(x) = A + B;b) limEx→a (f · g)(x) = A · B;Ac) limEx→a ( fg ) = B, if B = 0 and g(x) = 0 for x ∈ E.As already noted at the beginning of Sect. 3.2.2, this theorem is an immediateconsequence of the corresponding theorem on limits of sequences, given Proposition 1.
The theorem can also be obtained by repeating the proof of the theorem onthe algebraic properties of the limit of a sequence. The changes needed in the proofin order to do this reduce to referring to some deleted neighborhood Ů E (a) of a inE, where previously we had referred to statements holding “from some N ∈ N on”.We advise the reader to verify this.Here we shall obtain the theorem from its simplest special case when A = B = 0.Of course assertion c) will then be excluded from consideration.A function f : E → R is said to be infinitesimal as E x → a iflimEx→a f (x) = 0.3.2 The Limit of a Function113Proposition 2 a) If α : E → R and β : E → R are infinitesimal functions as E x → a, then their sum α + β : E → R is also infinitesimal as E x → a.b) If α : E → R and β : E → R are infinitesimal functions as E x → a, thentheir product α · β : E → R is also infinitesimal as E x → a.c) If α : E → R is infinitesimal as E x → a and β : E → R is ultimatelybounded as E x → a, then the product α · β : E → R is infinitesimal as E x → a.Proof a) We shall verify that lim α(x) = 0 ∧ lim β(x) = 0 ⇒ lim (α + β)(x) = 0 .Ex→aEx→aEx→aLet ε > 0 be given.
By definition of the limit, we have εlim α(x) = 0 ⇒ ∃Ů E (a) ∀x ∈ Ů E (a) α(x) <,Ex→a2 ε.lim β(x) = 0 ⇒ ∃Ů E (a) ∀x ∈ Ů E (a) β(x) <Ex→a2Then for the deleted neighborhood Ů E (a) ⊂ Ů E (a) ∩ Ů E (a) we obtain ∀x ∈ Ů E (a) (α + β)(x) = α(x) + β(x) ≤ α(x) + β(x) < ε.That is, we have verified that limEx→a (α + β)(x) = 0.b) This assertion is a special case of assertion c), since every function that has alimit is ultimately bounded.c) We shall verify that lim α(x) = 0 ∧ ∃M ∈ R ∃Ů E (a) ∀x ∈ Ů E (a) β(x) < M ⇒Ex→a⇒lim α(x)β(x) = 0 .Ex→aLet ε > 0 be given.
By definition of limit we have εlim α(x) = 0 ⇒ ∃Ů E (a) ∀x ∈ Ů E (a) α(x) <.Ex→aMThen for the deleted neighborhood Ů E (a) ⊂ Ů E (a) ∩ Ů E (a), we obtain ε· M = ε.∀x ∈ Ů E (a) (α · β)(x) = α(x)β(x) = α(x)β(x) <MThus we have verified that limEx→a α(x)β(x) = 0.The following remark is very useful:1143LimitsRemark 1 lim f (x) = A ⇔ f (x) = A + α(x) ∧ lim α(x) = 0 .Ex→aEx→aIn other words, the function f : E → R tends to A if and only if it can be represented as a sum A + α(x), where α(x) is infinitesimal as E x → a.
(The functionα(x) is the deviation of f (x) from A.)10This remark follows immediately from the definition of limit, by virtue of whichlim f (x) = A ⇔ lim f (x) − A = 0.Ex→aEx→aWe now give the proof of the theorem on the arithmetic properties of the limitof a function, based on this remark and the properties of infinitesimal functions thatwe have established.Proof a) If limEx→α f (x) = A and limEx→a g(x) = B, then f (x) = A + α(x)and g(x) = B + β(x), where α(x) and β(x) are infinitesimal as E x → a. Then(f + g)(x) = f (x) + g(x) = A + α(x) + B + β(x) = (A + B) + γ (x), where γ (x) =α(x) + β(x), being the sum of two infinitesimals, is infinitesimal as E x → a.Thus limEx→α (f + g)(x) = A + B.b) Again representing f (x) and g(x) in the form f (x) = A + α(x), g(x) =B + β(x), we have(f · g)(x) = f (x)g(x) = A + α(x) B + β(x) = A · B + γ (x),where γ (x) = Aβ(x) + Bα(x) + α(x)β(x) is infinitesimal as E x → a becauseof the properties just proved for such functions.Thus, limEx→a (f · g)(x) = A · B.c) We once again write f (x) = A + α(x) and g(x) = B + β(x), wherelimEx→a α(x) = 0 and limEx→a β(x) = 0.Since B = 0, there exists a deleted neighborhood Ů E (a), at all points of which|B||β(x)| < |B|2 , and hence |g(x)| = |B + β(x)| ≥ |B| − |β(x)| > 2 .
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