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

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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 ε > 0 there exists δ > 0 such that for all x < a with 0 < | x a | < δ we have | f ( x ) – L | < ε . 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 ε > 0 there exists δ > 0 such that for all x > a with 0 < | x a | < δ we have | f ( x ) – L | < ε . 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 δ 0 such that f ( x ) = g ( x ) for all x satisfying 0 < | x a | < δ 0 ) then lim x a f ( x ) = lim x a g ( x ).

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

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