For each natural number

n

and each number

x

in

(-1,1)

, define\

f_(n)(x)=\\\\sqrt(x^(2)+(1)/(n))

\ and define

f(x)=|x|

. Prove that the sequence

{f_(n)}

converges uniformly on the\ open interval

(-1,1)

to the function

f

. Check that each function

f_(n)

is continuously\ differentiable, whereas the limit function

f

is not differentiable at

x=0

. Does this\ contradict Theorem 9.33?

1. For each natural number n and each number x in (1,1), define fn(x)=x2+n1 and define f(x)=x. Prove that the sequence {fn} converges uniformly on the open interval (1,1) to the function f. Check that each function fn is continuously differentiable, whereas the limit function f is not differentiable at x=0. Does this contradict Theorem 9.33