The value of $\lim\limits_{n\to \infty}x\bigg[\tan^{-1}\large\frac{x+1}{x+2}$$-\cot^{-1}\large\frac{x+2}{x}\bigg]$ is

Question

$(a)\;-1\qquad(b)\;\large\frac{-1}{2}$$\qquad(c)\;\large\frac{1}{2}$$\qquad(d)\;1$