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# assignment 3 answers - Section 2.4 Page 1 Section 2.5 Page...

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Math 151 - D100 Fall 2010, Solutions to instructor problems HW3 A.1 For which a, b R is the function f ( x ) = 1 - x - 1 ax if x (0 , 1] 1 if x = 0 bx 4 + bx x 2 + x if x ( - 1 , 0) continuous on ( - 1 , 1]? Using the theorems in section 2.5 of the textbook that state: 1) that rational func- tions are continuous everywhere on their domain, 2) that root functions are con- tinuous at every number in their domain, and 3) that the sum of two continuous functions is continuous, we can state that: 1. 1 - x - 1 ax is continuous on (0 , 1] 2. bx 4 + bx x 2 + x is continuous on ( - 1 , 0) It remains to be shown then that f ( x ) is continuous at x = 0. By the definition of continuity, for f ( x ) to be continuous at x = 0, we must have that lim x 0 of f ( x ) = f (0). f (0) = 1 by the definition of f ( x ). Therefore, the limit of f ( x ) as x approaches 0 must be 1. This in turn means that both the left hand and right hand limit of f ( x ) as x approaches 0 must be 1. The statement about the left hand limit means that lim x 0 - bx 4 + bx x 2 + x = 1 or lim x 0 - bx ( x 3 + 1) x ( x + 1) = lim x 0 - b ( x 3 + 1) ( x + 1) = 1 Therefore

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