9.1 Families of Functions and Pointwise Convergence 547- For each of the following sequences of functions Un}, find the largest set
S on which the sequence {f n} converges pointwise, and find the limit
function f on that set. Sketch graphs of these functions where practical.
x
(a) fn(x) = -
n
(c) fn(x) = nxe-nx
(e) fn(x) = sin^2 n x
nx
(g) fn(x) = 1 + nx
xn
(i) fn(x) = --
1 +xn
x 2 -nx
(k) f n ( x) = ne ~ n ~
n +(b) fn(x) = tan-^1 (nx)
(d) fn(x) = (1 - lxl)n(f) fn(x) = cosn(x)
nx
(h) fn(x) = 1 + n2x2
2
(j) fn(x) = x3n++ n;(1) fn(x) = (1 + ~r
- Despite the negative tone of Examples 9.1.7, some properties of functions
are preserved by pointwise limits of sequences. For example, suppose {f n}
is a sequence of functions defined on a set S, converging pointwise to a
limit function f on S. Prove that
(a) if each function fn is bounded above (or below) by the same constant
Mon S , then so is f.
(b) if each function f n is monotone increasing (or decreasing) on S, then
so is f. - Can you find a sequence Un} of functions, each of which has a local
maximum at some point x 0 but the limit function does not? - In a slight modification of Example 9.1.7 (b), define fn: [O, 1]---+ JR by
{0 if x = 0,
fn(x) = 2n - 2n
2
x if 0 < x ~ *'
0 if * < x ~ 1,
and let f be the pointwise limit of f on [O, l]. Show that f n and f are
integrable on [O, 1] but lim f 0
1
f n -/:-f 0
1
n->oo f.- Define fn : [O, 1] ---+JR by
f n ( X) = { 3n; 1 ~!^0 < x ~ *'
x = 0 or x > *'
and let f be the pointwise limit off on [O, l]. Show that fn and f are
integrable on [O, 1] but lim f 01
f n -/:-f 01
n->oo f.