Calculating $\lim_{n\rightarrow\infty} \sum_{x = 1}^{n}{\frac{x}{n^2+x^2} }$

Question

Is there a simple method of calculating this limit?

$\lim_{n\rightarrow\infty} \sum_{x = 1}^{n}{\frac{x}{n^2+x^2} }.$

Answer 1

Write x/(x^2 + n^2) = 1/n*[(x/n)/(1 + (x/n)^2))] . So Lim S(x,n) = Int( 0 --> 1)(x/(1 + x^2)dx) = 1/2*Ln(1 + x^2) evaluated at x = 0 and x = 1. So Lim S(x,n) = ln2/2.