$\lim\limits_{n\to \infty}\sum_{i=1}^{n}\sum_{j=1}^{n}\frac{n}{(n+i)(n^2+j^2)}$
the form of limits reminds me $\iint\limits_{D} f(x,y)dxdy = \lim\limits_{\lambda\to 0}\sum_{i=1}^{\infty}\sum_{j=1}^{\infty}f(x,y)dxdy$ $D=\{(x,y)|x\in [1,\infty],y\in [1,\infty]\}$
but that is all I can think about this problem
$$ \lim\limits_{n\to \infty }\sum_{i=1}^{\infty}\sum_{j=1}^{\infty}\frac{n}{(n+i)(n^2+j^2)} = \lim\limits_{n\to \infty }\sum_{i=1}^{\infty}\sum_{j=1}^{\infty}\left(\frac{1}{1+\frac{i}{n}}\right)\frac{1}{n}\left(\frac{1}{1+(\frac{j}{n})^2}\right)\frac{1}{n} $$
and finally
$$ \lim_{n\to \infty}\sum_{i=1}^{\infty}\left(\frac{1}{1+\frac{i}{n}}\right)\frac{1}{n}\sum_{j=1}^{\infty}\left(\frac{1}{1+(\frac{j}{n})^2}\right)\frac{1}{n} = \int_0^1\frac{d \xi}{1+\xi}\int_0^1\frac{d\eta}{1+\eta^2}=\frac{\pi}{4}\ln (2) $$