Question

The function f(x) =x/(1 + x^2) has the x-axis as a horizontal asymptote. Indeed, f(x) =1/ (1/x + x)-> 0 as x -> infinity.

So, we expect that for every epsilon > 0 there is a large number K such that |f(x)| < epsilon when x > K. For epsilon > 0, find a suitable such K, and show that it is suitable.