Find the limits of the following sequences of functions Dete

Find the limits of the following sequences of functions. Determine whether the uniform. f_n (x) = {0, x = 0 x, 0

Solution

1) The function converge uniformly

The function is converging for n=0, since the value of function is equal to 0

The function will tend to zero, f(0<x<1/n) = x, since n will be increasing in the poiintwise function, so with the increase function will value also tend to zero

The function x^2 will tend to zero, since 1/n will tend to zero as n will tend to infinity, hence the function is uniform

2) The function is uniform since the maximum value of the function hn(x) will be 1, when x=1 and n is tending to infinity

3) The function is not uniform, since x can be equal to infinity and n will also tend to infinity will make the limit of pointwise function equal to infinity