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**Unformatted text preview: **The Epsilon-Delta Definition of Limit of a Function David Radford 10/06/05 Suppose that f : R- R is a function and a,L R . Then lim x- a f ( x ) = L means for all positive real numbers there exists a positive real number such that 0 < | x- a | < implies | f ( x )- L | < . This is the epsilondelta definition of the limit of the function y = f ( x ) at x = a . We can reformulate the definition in terms of quantifiers as > , ( > , ( x ( a- ,a + )- { a } , | f ( x )- L | < )) . Usually > 0 is shorthand for R + and > 0 is shorthand for R + . We note that the definition of limit of function is more complicated when the domain of f ( x ) is not all of R . The negation of a A,P ( a ) is not ( a A,P ( a )) which is equivalent to a A, not P ( a ) . The negation of a A,P ( a ) is not ( a A,P ( a )) which is equivalent to a A, not P ( a ) ....

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