Let for $n\geq 1$: $f_n(x):=\dfrac{x}{1+nx^2}$ and $\mathcal{F}:=\{f_n:n=1,2,3,\ldots\}$
I'd like to know if $\mathcal{F}$ is equicontinuous.
This is what I have done:
$\left|\frac{x}{1+nx^2}-\frac{x_0}{1+nx_0^2}\right|<\varepsilon$
this becames $|x-x_0|\left|\frac{1-nxx_0}{(1+nx^2)(1+nx_0^2)}\right|<\varepsilon$. Now I thought to define this function $g(n)=\frac{1-nxx_0}{(1+nx^2)(1+nx_0^2)}$ where we think $n\in\mathbb{R}$ and $x,x_0$ as parameters. Now $f(n)\rightarrow0$ if $n\rightarrow\infty$ and it is continuous, so there will be an interval $[-M,M]$ such that this function will be less than a certain $\nu$ ($\nu$ and $M$ depend on $x$ and $x_0$). And on $[-M,M]$ $g(n)$ will have a maximum $N$ (depending on $x,x_0$). So we have
$|x-x_0|\left|\frac{1-nxx_0}{(1+nx^2)(1+nx_0^2)}\right|<\delta\cdot\mathrm{max}\{{N,\nu}\}$
But now I don't know how to continue, please could you help me?