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→∞