undersetn→∞limΣΣlimits_1≤i<j≤n((i+1)(j+1))/(n^4) is

$\underset{n→∞}{\lim}\frac{\left(\sum\limits_{i=1}^{n}(i+1)\right)^2=\sum\limits_{i=1}^{n}(i+1)^2}{2n^4}$

$=\underset{n→∞}{\lim}\frac{\left(\frac{(n+2)(n+1)}{2}-1\right)^2-\left(\frac{(n+1)(n+2)(2n+3)}{6}\right)}{2n^4}=\frac{1}{8}$