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Unformatted text preview: Problem 1 (10 points) . This problem concerns various limits and deriva- tives. If the limit does not exist, then please so state. lim x + x 2 + 1 x 2 + 1 = lim x + 1 x 2 + 1 = 0 lim x sin x x = 1 lim x 1- x 2- 1 x 2- 2 x + 1 = lim x 1- ( x- 1)( x + 1) ( x- 1) 2 = lim x 1- x + 1 x- 1 =- lim x + sin x cos x = The function is periodic and does not have any limit at + . d dx x x 2 + 1 = 1 2 x ( x 2 + 1)- x (2 x ) ( x 2 + 1) 2 d dx (tan(sin x )) = cos x cos 2 (sin x ) = sec 2 (sin x ) cos x When x > 1 , d dx tan- 1 p x 2- 1 = 2 x 2 x 2- 1 1 + x 2- 1 = 1 x x 2- 1 d dx ( sin- 1 ( x 2 ) ) = 2 x 1- x 4 d dx e x 2 / 2 = xe x 2 / 2 d dx ( ln( x 2 + 1) ) = 2 x x 2 + 1 In this problem simplification is not strictly required. 1 2 Problem 2 (2 points each.) . This problem concerns the precise statements of definitions and theorems concerning to limit, continuity, and differentiability. (1) Write the precise definition of lim x c f ( x ) = L using and . For every positive > 0, there is a positive > 0 such that whenever < | x- c | < , we have | f ( x )- L | < . (2) Give the definition for a function f ( x ) to be differentiable at an interior point x = c of its domain. A function f ( x ) is said to be differentiable at x = c if the limit lim h f ( c + h )- f ( c ) h exists. (3) Give the precise statement of the Chain Rule. Let y = f ( u ) be a function in u , and u = g ( x ) a function in x . If f ( u ) is differentiable at a point u = g ( x ) and the function g ( x ) is differentiable at a point x , then the composite function y = f ( g ( x ) ) is differentiable at x , and the derivative is given by...
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