PracticePrelim1Solutions-1

PracticePrelim1Solutions-1 - Math 1110 Practice Prelim 1...

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Math 1110 Practice Prelim 1 Solutions 1. Evaluate the following limits if they exist. If the limits do not exist say why they do not exist. a. lim x 2 x 2 +3 x x b. lim x 0 - 1 x 3 c. lim x →- 3 x 2 - x - 12 x +3 d. lim x →∞ 4 x 2 +2 x +1 3 x Solution: a. lim x 2 x 2 +3 x x = 2 2 +3 · 2 2 = 5. b. If x is very small and negative, then x 3 is also very small and negative and so 1 x 3 is negative and has large absolute value. So lim x 0 - 1 x 3 = -∞ . c. lim x →- 3 x 2 - x - 12 x +3 = lim x →- 3 ( x - 4)( x +3) x +3 = lim x →- 3 x - 4 = - 7. d. lim x →∞ 4 x 2 +2 x +1 3 x = lim x →∞ x q 4+ 2 x + 1 x 2 3 x = lim x →∞ q 4+ 2 x + 1 x 2 3 = 4 3 = 2 3 2. Determine the values of A and B so that the function f as defined below is continuous for all x . f ( x ) = Ax - B if x 1 x 2 + Bx - A if 1 < x 2 2 if x > 2 Solution: f is continuous for x < 1, 1 < x < 2 and x > 2 because it coincides with poly- nomials (which are continuous) on those open intervals. All we have to check is that f is continuous at x = 1 and x = 2.
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