Limit Proof (Find Sufficiently Large N)

Question

Fix $x \in \mathbb{R}$. I want to find $N>0$ such that $$ \Big| \frac{nx}{1+n^2x^2} \Big| < \epsilon $$ for all $n > N$.

I'm thinking there might be some clever algebraic trick to simplifying things, since this takes the form $$ \frac{a}{x^2 + a^2} $$ but nothing comes to mind. Help would be appreciated, thank you.

Answer 1

I don't think this can hold as for any $n$, you can always choose $x = 1/n$ so that $|\frac{nx}{1+n^2x^2}| = 1/2$

(x is fixed now after the update of the problem)

Trivial for $x = 0$.

Otherwise, $|\frac{nx}{1+n^2x^2}| = |\frac{1}{nx + 1/nx}| = \frac{1}{n|x| + 1/n|x|} \leq \frac{1}{n|x|}$