MHB How can I find a sequence of functions satisfying certain properties?

[evinda]

Hey! ;) I am looking at the following exercise:
Find a sequence of differentiable functions $f_n$,such that $f_n \to f$ uniformly,where $f$ is differentiable, $f_n' \to g$ pointwise,but $f'\neq g$.

How can I find such a sequence of functions? Is there a methodology to do it??

 

[Evgeny.Makarov]

One method is to look in the book "Counterexamples in Analysis". Consider $f_n(x)=x/(1+n^2x^2)$. Check the required properties.

 

[evinda]

One method is to look in the book "Counterexamples in Analysis". Consider $f_n(x)=x/(1+n^2x^2)$. Check the required properties.

I saw the solution of the textbook,but I didn't know how they found the sequence of functions $f_n$..