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Unformatted text preview: 92.131 Lecture 8 1 of 9 Ronald Brent © 2009 All rights reserved. Basic Idea of the Limit We say L x f a x = → ) ( lim if the function f can be made arbitrarily close to L by making the number x close to a . In terms of inputs and outputs: L x f a x = → ) ( lim means: As inputs to f approach a , outputs from f approach L . The most crucial thing about a limit is that the function need not be defined at the point a x = . This is always the case for the definition of the derivative. Example: Determine 1 1 lim 2 1 − − → x x x . This really is the limit ) ( lim 1 x f x → , where 1 1 ) ( 2 − − = x x x f . There are several ways to examine this. Table: x 0.9 0.99 0.999 1.001 1.01 1.1 ) ( x f 1.9 1.99 1.999 2.001 2.01 2.1 92.131 Lecture 8 2 of 9 Ronald Brent © 2009 All rights reserved. x54321 1 2 3 4 554321 1 2 3 4 y 5 Graph: Algebraic Simplification: Note that 1 ) 1 ( ) 1 )( 1 ( 1 1 ) ( 2 + = − − + = − − = x x x x x x x f , as long as 1 ≠ x . So that 2 ) 1 ( lim ) ( lim 1 1 = + = → → x x f x x . 92.131 92....
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This note was uploaded on 02/13/2012 for the course MATH 92.131 taught by Professor Staff during the Fall '09 term at UMass Lowell.
 Fall '09
 Staff
 Calculus

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