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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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This note was uploaded on 01/19/2009 for the course MAT 1214 taught by Professor Johnrayko during the Spring '08 term at The University of Texas at San Antonio- San Antonio.
- Spring '08