2. (9 points) For each sequence of functions {fn} find the pointwise limit (if possible). Determine whether or not the sequence of functions converges uniformly. Prove your claim on uniform convergence. (a) (2 points) fn: [0, infinity] - > R with fn(x) = cos(root n^2 + x)/n^4. (b)(2points)fn: [0,infinity] - > R given by fn(x) = x^n/1+x^n. (c) (2 points) fn: [0,infinity] - > R with fn(x) = x/ne^-x/n. (d) (3 points) fn: [0,infinity] - > R with fn(x) = n^2x(1 - x)^n. Solution a) As cos of any value is finite within -1 and +1, fnx converges to 0 b_)Leadin terms ratio is 1. Hence sequence converges to1 c) x/n e^(-x/n) = x/n(1-x/n+x^2/n^2 2! ..... tends to 0 d) n2x(1-x)n Substitute x =1-y we have n2(1-y)yn for y in (0,1) AS y is less than 1, yn tends to 0 faster than increase of n^2,. Hence sequence tends to 0.

1. 2. (9 points) For each sequence of functions {fn} find the pointwise limit (if possible).
Determine whether or not the sequence of functions converges uniformly. Prove your claim on
uniform convergence. (a) (2 points) fn: [0, infinity] - > R with fn(x) = cos(root n^2 + x)/n^4.
(b)(2points)fn: [0,infinity] - > R given by fn(x) = x^n/1+x^n. (c) (2 points) fn: [0,infinity] - > R
with fn(x) = x/ne^-x/n. (d) (3 points) fn: [0,infinity] - > R with fn(x) = n^2x(1 - x)^n.
Solution
a) As cos of any value is finite within -1 and +1, fnx converges to 0
b_)Leadin terms ratio is 1. Hence sequence converges to1
c) x/n e^(-x/n) = x/n(1-x/n+x^2/n^2 2! .....
tends to 0
d) n2x(1-x)n
Substitute x =1-y
we have n2(1-y)yn for y in (0,1)
AS y is less than 1, yn tends to 0 faster than increase of n^2,.
Hence sequence tends to 0