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Unformatted text preview: Math 124 Quiz 2 Sept. 8, 2006 Name: 1. Show, using the rigorous definition of continuity, that the following piece-wise function is not continuous at x = 0: g ( x ) = x | x | , x 6 = 0; , x = 0 Solution: Note that if x < 0 then x | x | = x- x =- 1 and if x > 0 then x | x | = x x = 1. Hence, lim x →- g ( x ) = lim x →- x | x | = lim x →- (- 1) =- 1 and lim x → + g ( x ) = lim x → + x | x | = lim x → + (1) = 1 This shows that lim x → g ( x ) does not exist, since the limit from the left does not equal the limit from the right. Now, if g were to be continuous at 0, it would be true that lim x → g ( x ) = g (0). However, since the limit on the left doesn’t even exist, this last equation has no hope of being true. We conclude that g is not continuous at 0. 2. Show mathematically that there is a number c with 0 ≤ c ≤ 1 such that f ( c ) = 0, where f ( x ) = 2 x- 1 x . Solution: This is a case for the famous Intermediate Value Theorem. First note that f is continuous. Sinceis continuous....
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