Here's what we tried:
For every $\epsilon > 0$ there is a large number $K$ such that $|f(x)| < \epsilon$ when $x>K$.
Knowing that $K$ is a large positive number, take the positive absolute value of $f(x)$:
$$\displaystyle \frac{x}{1+x^2} < \epsilon$$
Solve for
$$x > \displaystyle \sqrt{\frac{x - \epsilon}{\epsilon}}$$
And thus $$K = \displaystyle \sqrt{\frac{x - \epsilon}{\epsilon}}$$.
Is it acceptable to have $K$ in terms of $x$?