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Math 447, Homework 8.∗ You may use standard calculus results (as applied to Riemann integrals) freely. 1. Compute the limit Z lim n→∞ 0 ∞ n sin(x/n) dx. x(1 + x2 ) Make sure to justify your calculation. 2. Exercise 18.39 (page 332). 3. Exercise 18.40 (page 332). 4. Exercise 18.41 (page 332). Hint: use the triangle inequality for one direction and an earlier exercise for the other. Quiz 8. This problem is about “differentiation under the integral sign.” Let f (x, y), 0 ≤ x, y ≤ 1, satisfy the following conditions: for each x, f (x, y) is an integrable function of y, and (∂f (x, y)/∂x) is a bounded function of (x, y). Show that (∂f (x, y)/∂x) is a measurable function of y for each x and Z 1 Z 1 ∂ d f (x, y) dy = f (x, y) dy. dx 0 0 ∂x (x,y) Hint: denote fn (y) = f (x+hn ,y)−f for appropriate hn . hn ∗ c 2016 by Michael Anshelevich. 1