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ANALYSIS: Master’s Comprehensive Exam January ’06 Instructions: Attempt all of the problems, showing work. Place at most one problem solution on a side for each sheet of paper turned in. Do not submit your scratch work. (1) If 0 < x < π , 2 prove that 2 sin x < < 1. π x (2) Suppose that f : [0, ∞) → R is continuous and limx→∞ f (x) = 1. Z 1 R −x (a) For 0 < R < ∞, evaluate lim e n f (x) dx n→∞ n 0 Z 1 ∞ −x (b) Show lim e n f (x) dx = 1 n→∞ n 0 (3) Let (fn ) denote a sequence of continuous functions on a domain D ⊂ R such that fn converges uniformly to the function f on D. Show that lim fn (xn ) = f (x), if lim xn = x and x ∈ D. n→∞ n→∞