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# Quiz2s - (b(5 points Let f be the function f x = x 2-1 |...

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MATH 1501 Quiz 2 August 31, 2009 No books or notes allowed. No laptop, graphic calculator or wireless devices allowed. Write clearly. Name:

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MATH 1501 Quiz 2 August 31, 2009 1. (a) (5 points) Let f be the function f ( x ) = ( x - 1) 2 | x - 1 | . Can you define f for x = 1 so that f is continuous at 1. ( Hint: check the right and left limit separatly.) Solution: For x > 1 we have f ( x ) = ( x - 1) x - 1 | x - 1 | = x - 1 while for x < 1 f ( x ) = ( x - 1) x - 1 | x - 1 | = - ( x - 1) so that lim x 1 + f ( x ) = lim x 1 - f ( x ) = 0 Thus setting f (1) = 0 we have that f ( x ) is continuous at 1. Page 1 of 2
MATH 1501 Quiz 2 August 31, 2009
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Unformatted text preview: (b) (5 points) Let f be the function f ( x ) = x 2-1 | x-1 | . Can you deﬁne f for x = 1 so that f is continuous at 1. Solution: For x > 1 we have f ( x ) = ( x + 1) x-1 | x-1 | = x + 1 while for x < 1 f ( x ) = ( x + 1) x-1 | x-1 | =-( x + 1) so that lim x → 1 + f ( x ) = 2 lim x → 1-f ( x ) =-2 Thus f ( x ) has a jump discontinuity at 1. There is no way to deﬁne f for x = 1 so that f is continuous at 1. Page 2 of 2...
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