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 $x$?