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# Homework 1 - Rehman(aar638 HW01 sachse(56620 This print-out...

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Rehman (aar638) – HW01 – sachse – (56620) 1 This print-out should have 20 questions. Multiple-choice questions may continue on the next column or page – find all choices before answering. 001 10.0 points Determine lim x → ∞ x 2 8 x + 7 6 + 5 x 3 x 2 . 1. limit = 2. limit = 1 3 correct 3. none of the other answers 4. limit = 0 5. limit = 1 6 Explanation: Dividing the numerator and denominator by x 2 we see that x 2 8 x + 7 6 + 5 x 3 x 2 = 1 8 x + 7 x 2 6 x 2 + 5 x 3 . On the other hand, lim x → ∞ 1 x = lim x → ∞ 1 x 2 = 0 . By Properties of limits, therefore, the limit = 1 3 . 002 10.0 points Let P ( x ) and Q ( x ) be polynomials. Deter- mine lim x → ∞ P ( x ) Q ( x ) when P ( x ) has degree 2 and Q ( x ) has degree 4. 1. limit = −∞ 2. limit = 3. limit = 0 correct 4. limit = 4 5. limit = 2 6. not enough information given Explanation: Since P has degree 2 and Q has degree 4, there exist a, b negationslash = 0 such that P ( x ) = ax 2 + R ( x ) , Q ( x ) = bx 4 + S ( x ) where R ( x ) , S ( x ) are polynomials such that deg( R ) < 2 , deg( S ) < 4. Thus P ( x ) Q ( x ) = ax 2 + R ( x ) bx 4 + S ( x ) = a x 2 + R ( x ) x 4 b + S ( x ) x 4 . On the other hand, lim x → ∞ 1 x 2 = lim x → ∞ R ( x ) x 4 = lim x → ∞ S ( x ) x 4 = 0 since deg( R ) , deg( S ) < 4. Consequently, by Properties of Limits, lim x → ∞ P ( x ) Q ( x ) = 0 . 003 10.0 points Determine if lim x → ∞ braceleftBig ln(7 + 4 x ) ln(2 + 5 x ) bracerightBig exists, and if it does find its value. 1. limit = ln 4 5 correct 2. limit = ln 5 4 3. limit does not exist

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Rehman (aar638) – HW01 – sachse – (56620) 2 4. limit = ln 7 2 5. limit = ln 2 7 Explanation: By properties of logs, ln(7 + 4 x ) ln(2 + 5 x ) = ln parenleftbigg 7 + 4 x 2 + 5 x parenrightbigg = ln parenleftbigg 7 /x + 4 2 /x + 5 parenrightbigg . But lim x → ∞ 7 /x + 4 2 /x + 5 = 4 5 . Consequently, the limit exists and limit = ln 4 5 . 004 10.0 points Find the value of lim x → ∞ parenleftbigg 4 e x + 5 e x 2 e x 3 e x parenrightbigg . 1. limit = 1 5 2. limit = 1 5 3. limit = 2 correct 4. limit = 2 5. limit = 1 2 6. limit = 1 2 Explanation: After division we see that 4 e x + 5 e x 2 e x 3 e x = 4 + 5 e 2 x 2 3 e 2 x . On the other hand, lim x → ∞ e ax = 0 for all a > 0. But then by properties of limits, lim x → ∞ 4 + 5 e 2 x 2 3 e 2 x = 2 .
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Homework 1 - Rehman(aar638 HW01 sachse(56620 This print-out...

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