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Hwk6Notes - The Epsilon-Delta Definition of Limit of a...

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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 ² there exists a positive real number δ such that 0 < | x- a | < δ implies | f ( x )- L | < ² . This is the epsilon–delta definition of the limit of the function y = f ( x ) at x = a . We can reformulate the definition in terms of quantifiers as ∀ ² > , ( ∃ δ > , ( ∀ x ∈ ( a- δ,a + δ )- { a } , | f ( x )- L | < ² )) . Usually ∀ ² > 0 is shorthand for ∀ ² ∈ R + and ∃ δ > 0 is shorthand for ∃ δ ∈ 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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