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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 > 0, such that | f ( x )- L | < , whenever x D satisfies 0 < | x- c | < . 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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