lec15 - Math 117 – 15 Limits of functions Definition 15.1...

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Unformatted text preview: Math 117 – May 24, 2011 15 Limits of functions Definition 15.1. Let f : D → R , and let c be an accumulation point of D . We say that a real number L is a limit of f at c , if for every > 0 there exists δ > 0, such that | f ( x )- L | < , whenever x ∈ D satisfies 0 < | x- c | < δ . We denote the limit by lim x → c f ( x ) = L . The above definition is, of course, equivalent to the following statement in terms of neighborhoods. Theorem 15.2. Let f : D → R and let c be an accumulation point of D . Then lim x → c f ( x ) = L iff for each neiborhood V of L there exists a deleted neighborhood U of c , such that f ( U ∩ D ) ⊆ V . One can also define the limit of a function in terms of limits of sequences, as the next theorem illustrates. Theorem 15.3. Let f : D → R and c be an accumulation point of D . Then lim x → c f ( x ) = L iff for every sequence ( s n ) in D converging to c and s n 6 = c for all n ∈ N , the sequence ( f ( s n ) ) converges to L ....
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This note was uploaded on 05/31/2011 for the course MATH 117 taught by Professor Akhmedov,a during the Spring '08 term at UCSB.

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