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Unformatted text preview: f ( x ) = l n x , then f ( x ) = 1/ x . Thus, f (1) = 1 . Now, we use this fact to express the number e as a limit. From the definition of a derivative as a limit, we have: THE NUMBER e AS A LIMIT (Section 3.6) 1 (1 ) (1) (1 ) (1) '(1) lim lim ln(1 ) ln1 1 lim lim ln(1 ) lim ln(1 ) h x x x x x f h f f x f f h x x x x x x ++= = += = + = + As f (1) = 1 , we have If we put n = 1/ x , then n as x + . So, an alternative expression for e is: 1 lim ln(1 ) 1 x x x + = 1/ 1 lim ln(1 ) 1 ln(1 ) 1 lim lim(1 ) x x x x x x x x e e e e x + + = = = = + THE NUMBER e AS A LIMIT (Section 3.6) 1 lim(1 ) x x e x = + 1 lim 1 n n e n = +...
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 Fall '08
 Noohi
 Calculus, Derivative

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