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Limits at infinity-solutions HW 6

Limits at infinity-solutions HW 6 - handa(nh5757 Limits at...

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handa (nh5757) – Limits at infinity – bormashenko – (54880) 1 This print-out should have 15 questions. Multiple-choice questions may continue on the next column or page – find all choices before answering. 001 10.0 points Determine if the limit lim x → ∞ 3 x + 2 x 2 - x + 4 exists, and if it does, find its value. 1. limit = 2 2. limit = 0 correct 3. limit = - 3 4. limit = 4 5. limit = 2 4 6. limit doesn’t exist Explanation: Dividing in the numerator and denominator by x 2 , the highest power, we see that 3 x + 2 x 2 - x + 4 = 3 x + 2 x 2 1 - 1 x + 4 x 2 . On the other hand, lim x → ∞ 1 x = lim x → ∞ 1 x 2 = 0 . By Properties of limits, therefore, the limit exists and limit = 0 . 002 10.0 points A certain function f is known to have the properties lim x → -∞ f ( x ) = 1 , lim x → ∞ f ( x ) = 2 . Determine if lim x 0+ 2 + 5 x 3 + f ( 1 x ) exists, and if it does, compute its value. 1. limit = 7 4 2. limit = 1 3. limit = 1 2 4. limit does not exist 5. limit = 2 5 correct Explanation: The properties of f ensure that lim x 0 - f 1 x = lim x → -∞ f ( x ) = 1 , while lim x 0+ f 1 x = lim x → ∞ f ( x ) = 2 . By properties of limits, therefore, lim x 0 - 2 + 5 x 3 + f ( 1 x ) = 1 2 , while lim x 0+ 2 + 5 x 3 + f ( 1 x ) = 2 5 . Consequently, lim x 0+ 2 + 5 x 3 + f ( 1 x ) = 2 5 . 003 10.0 points Find all asymptotes of the graph of y = 3 x 2 - 10 x + 3 3 x 2 - 11 x + 6 .
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handa (nh5757) – Limits at infinity – bormashenko – (54880) 2 1. vertical: x = 2 3 , 3 , horizontal: y = 1 2. vertical: x = 2 3 , horizontal: y = 1 cor- rect 3. vertical: x = 3 , horizontal: y = 1 4. vertical: x = 2 3 , 3 , horizontal: y = - 1 5. vertical: x = 3 , horizontal: y = - 1 6. vertical: x = 2 3 , horizontal: y = - 1 7. vertical: x = - 2 3 , horizontal: y = 1 Explanation: After factorization y = (3 x - 1)( x - 3) (3 x - 2)( x - 3) . Thus y is not defined at x = 3, but for x = 3 y = 3 x - 1 3 x - 2 ; notice, however, that lim x 3 3 x - 1 3 x - 2 = 8 7 exists, so the graph does not have a vertical asymptote at x = 3. Since 3 x - 1 3 x - 2 -→ ∞ as x 2 3 from the left and the right, the line x = 2 3 will, however, be a vertical asymptote.
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