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f09exam1solutions

f09exam1solutions - Math 113 Exam#1 Solutions 1 What are...

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Math 113 Exam #1 Solutions 1. What are the domain and range of the function f ( x ) = 1 4 - x ? Answer: f ( x ) is well-defined provided 4 - x ) = 0. In order for 4 - x to exist, we must have 4 - x 0 , meaning that x 4. In order for 4 - x = 0 it must also be the case that x = 4, so the domain of f is ( -∞ , 4) . Since 4 - x > 0 where f is defined, we see that f ( x ) > 0 for all x in the domain, so the range of f is (0 , + ) . 2. Evaluate lim x →∞ 4 x 2 - 8 x + 7 17 x + 12 Answer: Dividing both numerator and denominator by x yields lim x →∞ 1 x 4 x 2 - 8 x + 7 1 x (17 x + 12) = lim x →∞ 1 x 2 (4 x 2 - 8 x + 7) 17 + 12 x = lim x →∞ 4 - 8 x + 7 x 2 17 + 12 x = 4 17 = 2 17 . 3. Let f ( x ) = x - 2 x 2 - 4 for x = 2 a for x = 2 If f ( x ) is continuous at x = 2, then find the value of a . Answer: In order for f to be continuous at x = 2, we must have that a = f (2) = lim x 2 f ( x ) = lim x 2 x - 2 x 2 - 4 . Now, we can factor the denominator in the limit to get lim x 2 x - 2 x 2 - 4 = lim x 2 x - 2 ( x + 2)( x - 2) = lim x 2 1 x + 2 = 1 4 , so we see that, in order for f to be continuous, a must be 1 4 .
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