Problem

Match the column.

Column I	Column II
(A) $\lim_{n \to \infty} \cos^2 (\pi (\sqrt[3]{n^3 + n^2 + 2n - n}))$ where $n$ is an integer, equals	(p) $\frac{1}{2}$
(B) $\lim_{n \to \infty} n \sin (2\pi \sqrt{1 + n^2})$ ( $n \in \mathbb{N}$ ) equals	(q) $\frac{1}{4}$
(C) $\lim_{n \to \infty} (-1)^n \sin (\pi \sqrt{n^2 + 0.5n + 1})$ $\left( \sin \frac{(n+1)\pi}{4n} \right)$ is (where $n \in \mathbb{N}$ )	(r) $\pi$
(D) If $\lim_{x \to \infty} \left( \frac{x+a}{x-a} \right)^x = e$ where $a$ is some real constant, the value of $a$ is equal to	(s) non-existent