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Unformatted text preview: (3/1908) Section 1.5 Formal definitions of limits Overview: The definitions of the various types of limits in previous sections involve phrases such as arbitrarily close, sufficiently close, arbitrarily large, and sufficiently large. For instance, according to Definition 2 of Section 1.1, lim x a f ( x ) = L with a number L if f ( x ) is arbitrarily close to L for all x negationslash = a sufficiently close to a . These qualitative formulations of the definitions are all that are needed in most of calculus. More quantitative formulations of the definitions are required, however, in dealing with difficult examples and in proofs. In these formal definitions , phrases such as arbitrarily close and sufficiently close are made precise by using inequalities. In this section we discuss the formal definition of two-sided finite limits lim x a f ( x ) = L . Formal definitions of lim x f ( x ) = L and lim x f ( x ) = are presented in the context of infinite sequences in Section 11.1. Other definitions are similar and are studied in advanced courses. Topics: The epsilon1-definition of a finite two-sided limit Using approximate inequalities Two short proofs Using graphs with finite limits The epsilon1-definition of a finite two-sided limit To convert Definition 2 in Section 1.1 of lim x a f ( x ) = L into a formal definition, we replace the phrase f ( x ) is arbitrarily close to L with | f ( x )- L | < epsilon1 for an arbitrarily small positive number epsilon1 , and we replace the phrase for all x negationslash = a sufficiently close to a with for all x with 0 < | x- a | < for a sufficiently small positive number . We obtain the following: Definition 1 (Finite two-sided limits) Suppose that y = f ( x ) is defined on open intervals ( b,a ) and ( a,c ) on both sides of a . Then lim x a f ( x ) = L with a number L if and only if for every positive epsilon1 , there is a positive such that | f ( x )- L | < epsilon1 for all x with 0 < | x- a | < . (1) epsilon1 and are the Greek letters epsilon and delta. 51 p. 52 (3/1908) Section 1.5, Formal definitions of limits Statement (1) means that the portions of the graph y = f ( x ) for a- < x < a and for a < x < a + are between the lines y = L + epsilon1 and y = L- epsilon1 as is illustrated in Figure 1. x y y = f ( x ) a + y = L + epsilon1 y = L- epsilon1 a epsilon1 epsilon1 L a- | f ( x )- L | < epsilon1 for < | x- a | < FIGURE 1 To see the geometric meaning of this Definition 1, imagine that the positive parameter epsilon1 in Figure 1 decreases toward zero, so that the lines y = L- epsilon1 and y = L + epsilon1 approach the line y = L . If the definition is satisfied, then for each epsilon1 , there is a such that the portion of the graph for a- < x < x on the left of a and the portion for a < x < a + on the right of a lie between those lines. Since epsilon1...
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