Calculus Pre-lim 1 w. Solutions

Calculus Pre-lim 1 w. Solutions - Math 111 Prelim 1 Name...

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Unformatted text preview: Math 111 Prelim 1 Sep 25, 2007 Name: Instructor: Section: INSTRUCTIONS — READ THIS NOW • This test has 8 problems on 11 pages worth a total of 100 points. Look over your test package right now . If you find any missing pages or problems please ask a proctor for another test booklet. • Write your name, your instructor’s name, and your section number right now . • Show your work. To receive full credit, your answers must be neatly written and logically organized. If you need more space, write on the back side of the preceding sheet, but be sure to label your work clearly. • This is a closed book exam. No calculators are allowed. You may bring a single 3x5 index card. • This exam is 90 minutes long. • Academic integrity is expected of all students of Cornell University at all times, whether in the presence or absence of members of the faculty. Understanding this, I declare I shall not give, use, or receive unauthorized aid in this examination. Signature of Student Math 111 (Fall 2007) Prelim 1 (Sep 25, 2007) 2 Question 1. (18 points) Evaluate the following limits, if they exist. If you use a theorem to help you get the answer, be sure to reference it in your solution. If a limit does not exist, indicate in what way it fails to exist. (For example, one sided limits don’t agree, or the function approaches + ∞ , or-∞ ). (a) (3 points) lim x → 2 − 1 √ 4- x 2 The limit does not exist because the function approaches + ∞ . (b) (3 points) lim x →∞ cos( x ) The limit does not exist because cos( x ) does not approach just one value as x → ∞ ; it oscillates between- 1 and 1 . (c) (3 points) lim x →− 4 2 x + 8 | x + 4 | The limit does not exist because the left and right limits do not agree: lim x →− 4 + 2 x + 8 | x + 4 | = lim x →− 4 + 2( x + 4) x + 4 = 2; lim x →− 4- 2 x + 8 | x + 4 | = lim x →− 4- 2( x + 4)- ( x + 4) =- 2 (d) (3 points) lim x → 1 parenleftbigg- 2 x 2- 1 + 1 x- 1 parenrightbigg lim x → 1 parenleftbigg- 2 x 2- 1 + 1 x- 1 parenrightbigg = lim x → 1 parenleftbigg- 2 ( x- 1)( x + 1) + x + 1 ( x- 1)( x + 1) parenrightbigg = lim x → 1 x- 1 ( x- 1)( x + 1) = lim x → 1 1 x + 1 = 1 2 Math 111 (Fall 2007) Prelim 1 (Sep 25, 2007) 3 (e) (3 points) lim x → bracketleftbigg sin 2 ( x ) cos parenleftbigg 1 x parenrightbiggbracketrightbigg- 1 ≤ cos parenleftbigg 1 x parenrightbigg ≤ 1 = ⇒ - sin 2 ( x ) ≤ sin 2 ( x ) cos parenleftbigg 1 x parenrightbigg ≤ sin 2 ( x ) Note that lim x → bracketleftbig- sin 2 ( x ) bracketrightbig...
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This note was uploaded on 04/07/2008 for the course MATH 1110 taught by Professor Martin,c. during the Spring '06 term at Cornell.

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Calculus Pre-lim 1 w. Solutions - Math 111 Prelim 1 Name...

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