12)
$\lim\limits_{n\to \infty}\dfrac{(n^2+3n+4-n^2-3n+4)[(n^2+3n+4)^2+(n^2+3n-4)^2+(n^2+3n+4)(n^2+3n-4)]}{ (n^2+5n+6-n^2-5n+6)[(n^2+5n+6)^2+(n^2+5n-6)^2+(n^2+5n+6)(n^2+5n-6)]}$
$=\lim\limits_{n\to \infty}\dfrac{2}{3}.\dfrac{(n^2+3n+4)^2+(n^2+3n-4)^2+(n^2+3n+4)(n^2+3n-4)}{(n^2+5n+6)^2+(n^2+5n-6)^2+(n^2+5n+6)(n^2+5n-6)}$
$=\lim\limits_{n\to \infty}\dfrac{2}{3}.\dfrac{ \Big(1+\dfrac{3}{n}+\dfrac{4}{n^2}\Big)^2+\Big(1+\dfrac{3}{n}-\dfrac{4}{n^2}\Big)^2+\Big(1+\dfrac{3}{n}-\dfrac{4}{n^2}\Big).\Big( 1+\dfrac{3}{n}-\dfrac{4}{n^2}\Big) }{ \Big( 1+\dfrac{5}{n}+\dfrac{6}{n^2}\Big)^2+\Big(1+\dfrac{5}{n}-\dfrac{6}{n^2}\Big)^2+\Big(1+\dfrac{5}{n}+\dfrac{6}{n^2}\Big).\Big( 1+\dfrac{5}{n}-\dfrac{6}{n^2}\Big)}$
$=\dfrac{2}{3}$
