09.26 - One-sided limits : Suppose f ( x ) is a function...

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Unformatted text preview: One-sided limits : Suppose f ( x ) is a function defined for all values of x close to but less than a ; that is, suppose there exists r > 0 such that f ( x ) is defined whenever x < a with | x a | < r . Then we say that lim x a f ( x ) = L iff there for every > 0 there exists > 0 such that for all x < a with 0 < | x a | < we have | f ( x ) L | < . Likewise: Suppose f ( x ) is a function defined for all values of x close to but greater than a ; that is, suppose there exists r such that f ( x ) is defined whenever x > a with | x a | < r . Then we say that lim x a + f ( x ) = L iff there for every > 0 there exists > 0 such that for all x > a with 0 < | x a | < we have | f ( x ) L | < . Theorem: lim x a f ( x ) = L if and only if lim x a f ( x ) and lim x a + f ( x ) are both defined and both equal L . Theorem: If f ( x ) and g ( x ) agree on some neighborhood of a (that is, if there exists 0 such that f ( x ) =...
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This note was uploaded on 02/13/2012 for the course MATH 141 taught by Professor Staff during the Fall '11 term at UMass Lowell.

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09.26 - One-sided limits : Suppose f ( x ) is a function...

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