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Sivakumar MATH 171 Exercise Set 3 From the text: Page Page Page Page 99: 15–19, 22, 26, 32, 34, 37, 38 100: 56, 64 101: 76, 78, 79 133: 4, 5, 6, 12, 13, 14–18, 24, 26, 28, 30 x2 − 1 . |x − 1| (i) Sketch the graph of F . (ii) Find lim− F (x). 1. Let F (x) = x→1 (iii) Find lim+ F (x). x→1 (iv) Does lim F (x) exist? Explain. x→1 2. Find numbers a and b such that x2 + ax − b = 2. x→a x−a lim The next pair of examples is taken from the book Basic Analysis: Japanese Grade 11, translated and published by the American Mathematical Society, as part of the University of Chicago School Mathematics Project. 3. Find the values of the constants a and b such that x2 − ax + 8 1 = . 2 x→2 x − (b + 2)x + 2b 5 lim 4. Find the cubic function f (x) (to wit, a function of the form f (x) = ax3 + bx2 + cx + d, where a, b, c, and d are constants) which satisfies the following conditions: f (x) =2 x→0 x lim and f (x) = 1. x→1 x − 1 lim 5. (i) Give a proof of the Sandwich Principle/Squeeze Theorem stated and discussed in lecture. (ii) Prove the following version of the Squeeze Theorem: Suppose that f (x) ≤ g(x) ≤ h(x) for every x > A (where A is some fixed real number). If lim f (x) = lim h(x) = L (where L is a x→∞ real number), then lim g(x) = L as well. x→∞ sin2 x . x→∞ x2 (iii) Use the theorem from part (ii) to evaluate lim 1 x→∞