Functions - PART III. FUNCTIONS: LIMITS AND CONTINUITY...

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PART III. FUNCTIONS: LIMITS AND CONTINUITY III.1. LIMITS OF FUNCTIONS This chapter is concerned with functions f : D R where D is a nonempty subset of R . That is, we will be considering real-valued functions of a real variable. The set D is called the domain of f . Defnition 1. Let f : D R and let c be an accumulation point of D . A number L is the limit o± f at c if to each ±> 0 there exists a δ> 0 such that | f ( x ) - L | whenever x D and 0 < | x - c | . This deFnition can be stated equivalently as follows: Defnition. Let f : D R and let c be an accumulation point of D . A number L is the limit o± f at c if to each neighborhood V of L there exists a deleted neighborhood U of c such that f ( U D ) V . Notation lim x c f ( x )= L . Examples: (a) lim x →- 2 ( x 2 - 2 x + 4) = 12. (b) lim x 2 x 2 - 4 x - 2 =4 . (c) lim x 3 x 2 +3 x +5 x - 3 does not exist. (d) lim x 1 | x - 1 | x - 1 does not exist. Example: Let f ( x )=4 x - 5. Prove that lim x 3 f ( x )=7 . Proo±: Let 0. | f ( x ) - 7 | = | (4 x - 5) - 7 | = | 4 x - 12 | | x - 3 | . Choose δ = ±/ 4. Then | f ( x ) - 7 | | x - 3 | < 4 ± 4 = ± whenever 0 < | x - 3 | . Two Obvious Limits: 23
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(a) For any constant k and any number c, lim x c k = k . (b) For any number c, lim x c x = c . THEOREM 1. Let f : D R and let c be an accumulation point of D . Then lim x c f ( x )= L if and only if for every sequence { s n } in D such that s n c, s n ± = c for all n , f ( s n ) L . Proof: Suppose that lim x c f ( x L . Let { s n } be a sequence in D which converges to c, s n ± = c for all n . Let ±> 0. There exists δ> 0 such that | f ( x ) - L | whenever 0 < | x - c | ( x D ) . Since s n c there exists a positive integer N such that | c - s n | for all n>N . Therefore | f ( s n ) - L | for all and f ( s n ) L. Now suppose that for every sequence { s n } in D which converges to c , f ( s n ) L . Suppose that lim x c f ( x ) ± = L . Then there exists an 0 such that for each 0 there is an x D with 0 < | x - c | but f ( x ) - L |≥ ± . In particular, for each positive integer n there is an s n D such that | c - s n | < 1 /n and | f ( s n ) - L ± . Now, s n c but { f ( s n ) } does not converge to L , a contradiction. Corollary Let f : D R and let c be an accumulation point of D . If lim x c f ( x ) exists, then it is unique. That is, f can have only one limit at c . THEOREM 2. f : D R and let c be an accumulation point of D .I f lim x c f ( x ) does not exist, then there exists a sequence { s n } in D such that s n c , but { f ( s n ) } does not converge. Proof: Suppose that lim x c f ( x ) does not exist. Suppose that for every sequence { s n } in D such that s n c ( s n ± = c ), { f ( s n ) } converges. Let { s n } and { t n } be sequences in D which converge to c . Then { f ( s n ) } and { f ( t n ) } are convergent sequences. Let { u n } be the sequence { s 1 ,t 1 ,s 2 2 ,... } . Then { u n }} converges to c and { f ( u n ) } converges to some number L . Since { f ( s n ) } and { f ( t n ) } are subsequences of { f ( u n ) } , f ( s n ) L and f ( t n ) L . Therefore, for every sequence { s n } in D such that s n c, s n ± = c for all n , f ( s n ) L and lim x c f ( x L .
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This note was uploaded on 01/06/2011 for the course MATH 3333 taught by Professor Staff during the Spring '08 term at University of Houston.

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Functions - PART III. FUNCTIONS: LIMITS AND CONTINUITY...

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