100_MT1-sample.soln - Math 100 105 Fall Term 2010 Sample...

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Unformatted text preview: Math 100 105, Fall Term 2010 Sample midterm Exam October 4 th , 2010 Student number: LAST name: First name: Instructions • Do not turn this page over. You will have 45 minutes for the exam. • You may not use books, notes or electronic devices of any kind. • Solutions should be written clearly, in complete English sentences, showing all your work. • If you are using a result from the textbook, the lectures or the problem sets, state it properly. Signature: 1 /18 2 /8 3 /6 4 /8 Total /40 1 Math 100 105, Fall 2010 Sample midterm, page 2/6 Problem 2 1 Short-form answers Show your work and clearly delineate your nal answer. Not all problems are of equal di culty. [3] a. Evaluate the following limit (or show it does not exist): lim x →- 2 x 2- 4 x 2 + 2 x + 1 = lim x →- 2 ( x 2- 4) lim x →- 2 ( x 2 + 2 x + 1) = 1 = 0 where the second equality holds since polynomials are continuous, and the rst holds by the quotient rule (since the limit of the denominator is non-zero). [3] b. Evaluate the following limit (or show it does not exist): lim x → e 3 x- 1 x Writing f ( x ) = e 3 x and noting that f (0) = 1 the given limit is simply the derivative f (0) . By the chain rule f ( x ) = 3 e 3 x so the given limit is 3 e = 0 . [3] c. Evaluate the following limit (or show it does not exist): lim x →∞ x cos x x 2 + 1 For any x we have- 1 ≤ cos x ≤ 1 and < 1 x 2 +1 ≤ 1 x 2 . Thus for x positive we have- 1 x =- x x 2 ≤ x cos x x 2 + 1 ≤ x x 2 =...
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