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119Mdt1Solutions

# 119Mdt1Solutions - METU Northern Cyprus Campus Calculus and...

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Last Name Name Student No.: Department: Section: Signature: : : : : : : : : : : Code Acad.Year Semester Date Time Duration M E T U Northern Cyprus Campus Calculus and Analytical Geometry I. Midterm Math 119 2007-2008 Fall 2.11.2007 17:40 120 minutes 6 QUESTIONS ON 6 PAGES TOTAL 101 POINTS 1 2 3 4 5 6 1. (4+4+4+4=16 points) Evaluate each of the following limits if it exists, or else show why it doesn’t exist. (Do not use L’Hospital’s rule.) (a) lim x 6 sin( x - 6) x 2 - 36 = lim x 6 sin( x - 6) x - 6 lim x 6 1 x + 6 = 1 12 (b) lim x 1 x 2 + 2 x - 3 x 2 - 3 x + 2 = lim x 1 ( x + 3)( x - 1) ( x - 2)( x - 1) = lim x 1 ( x + 3) ( x - 2) = - 4 (c) lim x 0 ( x | x | + x | x | ) lim x 0 - ( x | x | + x | x | ) = lim x 0 - ( - x 2 + x - x ) = - 1 lim x 0 + ( x | x | + x | x | ) = lim x 0 + ( x 2 + x x ) = 1 Since the left and right limits do not agree, the limit does not exist. (d) lim x + ( x 6 + 5 x 3 - x 3 ) = lim x + ( x 6 + 5 x 3 - x 3 ) ( x 6 + 5 x 3 + x 3 ) ( x 6 + 5 x 3 + x 3 ) = lim x + x 6 + 5 x 3 - x 6 ( x 6 + 5 x 3 + x 3 ) = lim x + 5 x 3 x 3 ( q 1 + 5 x 3 + 1) = 5 2

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2. (a) (10 points) Using the ² - δ definition of limit, prove that lim x 3 (4 x - 10) = 2 We want to show that for all ² > 0 there exists δ >
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119Mdt1Solutions - METU Northern Cyprus Campus Calculus and...

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