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Unformatted text preview: The Precise Definition of a Limit We wish to give precise meanings to statements such as “ x is close to a ”, and to give a precise definition of lim x → a f ( x ) = L . The distance between two numbers a and b is given by  a b  . For example, the distance between 4 and 7 is  4 7  = 3. So to say two numbers x and a are close, we ask that  x a  be small. We can say  x 3  < . 01 to say that x is within 0 . 01 of 3. We often use the Greek letters δ and ² (delta and epsilon) to denote small positive numbers. We might say “let δ > 0 and suppose  x a  < δ ”, to say that x is to be close to a . We might say “let ² > 0 and suppose  f ( x ) L  < ² ”, to say that f ( x ) is close to L . Example. Consider lim x → 3 (4 x 7) = 5. We ask the question: how close must x be to 3 for 4 x 7 to be within 0 . 01 units of 5? For 4 x 7 to be within 0 . 01 units of 5, we must have  (4 x 7) 5  < . 01. We can solve this inequality. We have  4 x 7 5  < . 01 or  4 x 12  < . 01 or  4( x 3)  < . 01. So 4  x 3 ...
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 Spring '08
 JohnRayko
 Calculus, Negative and nonnegative numbers, L. Example

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