The function f(x) =x/(1 + x^2) has the x-axis as a horizontal asymptote. So, we expect that for every epsilon 0 there is a large number K such that |f(x)| K. For epsilon 0, ﬁnd a suitable such K, and show that it is suitable.

Question

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

jamesj · Answer

1 + x^2 > x^2
Hence
1/x^2 > 1/(1+x^2)

Thus$$\left| {x \over {1+x^2}} \right| \leq \left| {x \over x^2} \right| = {1 \over |x|}$$

Hence given epsilon e, choose K = 1/e.  Then
|x| > K => 1/|x| < 1/K = e
            => ....