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.
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