Let $f_n(x) =\frac{x}{1+nx^2}$ and what function does this sequence converge to?

Question

Let $f_n(x) =\frac{x}{1+nx^2}$

I think that this sequence of function converges to 0.

And compute $f'_n(x)$ and find all values of $x$ for which $f'(x)=\lim f'_n(x)$

I found $f'_n(x)=\left(\frac{1-nx^2}{1+nx^2}\right)^2$ then it seems like that $\lim f'_n(x)=0$ for all $x$ except $x=0$.

Is it correct?

Answer 1

Hint:

$f_n(x)=\frac{x}{1+nx^2}\le \frac{x}{nx^2}=\frac{1}{nx}$, $x\not= 0$