Consider the sequence of functions f_n (x)= x / (1+nx^2)

let f = lim f_n. Compute f'_n and find all values of x such that f'(x)=lim f'_n (x)

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- November 16th 2011, 05:53 PMveronicak5678Uniform Convergence
Consider the sequence of functions f_n (x)= x / (1+nx^2)

let f = lim f_n. Compute f'_n and find all values of x such that f'(x)=lim f'_n (x) - November 16th 2011, 06:33 PMTKHunnyRe: Uniform Convergence
Can you state 'f' explicitly? Is it constant, linear, what?

Can you find f'_n, generally? - November 16th 2011, 06:45 PMveronicak5678Re: Uniform Convergence
f_'n (x)=(1-nx^2)/(1+nx^2 )^2 and f=0

- November 17th 2011, 05:15 AMTKHunnyRe: Uniform Convergence
Very good. You're almost there. Where is the derivative zero?

- November 17th 2011, 08:49 AMveronicak5678Re: Uniform Convergence
The derivative is zero when nx^2 =1, or when x=(1/n)^1/2

But isn't lim f' =0 for all x? - November 17th 2011, 10:33 AMTKHunnyRe: Uniform Convergence
Let's try to remember which variable is moving. Are we talking about x approaching zero or n increasing without bound?

AT x = 0, what is the limit of f'_n as n increases without bound? - November 17th 2011, 02:06 PMveronicak5678Re: Uniform Convergence
At x=0, f'_n =1

- November 17th 2011, 02:44 PMTKHunnyRe: Uniform Convergence
Right, and that's not zero (0), is it?

I believe you were done with this problem when you said this: "x=(1/n)^(1/2)" - November 17th 2011, 02:49 PMveronicak5678Re: Uniform Convergence
I guess I thought there was more to it. Thanks for your help.