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M408K Hm. 10 Solutions

# M408K Hm. 10 Solutions - Mahon Kevin Homework 10 Due Nov 3...

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Mahon, Kevin – Homework 10 – Due: Nov 3 2005, 3:00 am – Inst: Edward Odell 1 This print-out should have 18 questions. Multiple-choice questions may continue on the next column or page – find all choices before answering. The due time is Central time. 001 (part 1 of 3) 10 points A certain function f is given by the graph 4 8 - 4 - 8 4 - 4 - 8 (i) What is the value of lim x → -∞ f ( x ) 1. limit = 2 2. limit = - 4 3. limit = 4 4. limit does not exist 5. limit = - 2 correct Explanation: To the left of x = - 2 the graph of f os- cillates about the line y = - 2 and as x ap- proaches -∞ the oscillations become smaller and smaller. Thus limit = - 2 . 002 (part 2 of 3) 10 points (ii) What is the value of lim x → ∞ f ( x )? 1. limit = - 2 2. limit = - 4 3. limit = 2 4. limit = 4 correct 5. limit does not exist Explanation: To the right of x = 1 the graph of f is asymptotic to the line y = 4. Thus limit = 4 . 003 (part 3 of 3) 10 points (iii) What is the value of lim x → - 2 f ( x )? 1. limit = correct 2. limit = 2 3. limit = 4 4. limit = - 2 5. limit = - 4 Explanation: From the graph of f the left hand limit lim x → - 2 - f ( x ) = ,

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Mahon, Kevin – Homework 10 – Due: Nov 3 2005, 3:00 am – Inst: Edward Odell 2 while the right hand limit lim x → - 2+ f ( x ) = . Thus the two-sided limit lim x → - 2 f ( x ) = . keywords: Stewart5e, 004 (part 1 of 1) 10 points Determine if the limit lim x → -∞ 2 x x - 1 + 6 x x + 1 exists, and if it does, find its value. 1. limit = 5 2. limit = 6 3. limit = 8 correct 4. limit = 4 5. limit = 7 6. limit doesn’t exist Explanation: Bringing the expression to a common de- nominator, we see that 2 x x - 1 + 6 x x + 1 = 2 x ( x + 1) + 6 x ( x - 1) ( x - 1)( x + 1) = 8 x 2 - 4 x x 2 - 1 . Thus after dividing through by x 2 we see that lim x →-∞ 2 x x - 1 + 6 x x + 1 = lim x →-∞ 8 - 4 x 1 - 1 x 2 . Consequently, the limit exists and lim x → -∞ 2 x x - 1 + 6 x x + 1 = 8 . keywords: Stewart5e, 005 (part 1 of 1) 10 points Determine if lim x → ∞ x p 9 x 2 + 4 - 3 x · exists, and if it does, find its value. 1. limit = 0 2. limit = 1 2 3. limit = 1 6 4. limit = 2 3 correct 5. limit does not exist 6. limit = 1 3 Explanation: After rationalization we see that x p 9 x 2 + 4 - 3 x · = x 9 x 2 + 4 - 9 x 2 9 x 2 + 4 + 3 x = 4 x 9 x 2 + 4 + 3 x = 4 r 9 + 4 x 2 + 3 . On the other hand, lim x → ∞ r 9 + 4 x 2 = 3 . Consequently, lim x → ∞ x p 9 x 2 + 4 - 3 x ·
Mahon, Kevin – Homework 10 – Due: Nov 3 2005, 3:00 am – Inst: Edward Odell 3 exists and has limit = 2 3 . keywords: Stewart5e, limit, limit at infinity, square root function, rationalize numerator 006 (part 1 of 1) 10 points A certain function f is known to have the properties lim x → -∞ f ( x ) = 4 , lim x → ∞ f ( x ) = 1 . Determine if lim x 0+ 2 + 5 x 5 + f ( 1 x ) exists, and if it does, compute its value. 1. limit = 7 9 2. limit does not exist 3. limit = 5 6 4. limit = 1 3 correct 5. limit = 2 9 Explanation: The properties of f ensure that lim x 0 - f 1 x = lim x → -∞ f ( x ) = 4 , while lim x 0+ f 1 x = lim x → ∞ f ( x ) = 1 .

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M408K Hm. 10 Solutions - Mahon Kevin Homework 10 Due Nov 3...

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