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Unformatted text preview: Math 230 E Fall 2011 The Definition of a Limit Drew Armstrong In class we discussed the universal ( ∀ ) and the existential ( ∃ ) quantifier, as well as the following general principle: NOT ∀ x,P ( x ) = ∃ x, NOT P ( x ) . In words, this says that the opposite of the statement “for all x , the prop- erty P ( x ) holds” is “there exists some x such that P ( x ) does not hold”. With this principle we can negate rather complicated expressions, such as ∀ x ∃ y P ( x )AND Q ( y ), by working from left to right: NOT( ∀ x, ∃ y, P ( x )AND Q ( y )) = ∃ x, NOT( ∃ y, P ( x )AND Q ( y )) = ∃ x, ∀ y, NOT( P ( x )AND Q ( y )) = ∃ x, ∀ y, (NOT P ( x ))OR(NOT Q ( y )) . Do such complicated expressions really occur “in nature”; that is to say, “in mathematics”? Yes! In fact, the whole subject of Calculus is based on the following (horrifying?) definition. Definition. Given a function f : R → R , we say that lim x → a f ( x ) = L if ∀ ε > , ∃ δ > , ∀...
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