# $\lim\limits_{x\to 1}\bigg[\large\frac{x^3+2x^2+x+1}{x^2+2x+3}\bigg]^{\Large\frac{1-\cos(x-1)}{(x-1)^2}}$

$(a)\;e\qquad(b)\;e^{\Large\frac{1}{2}}\qquad(c)\;1\qquad(d)\;\sqrt{\large\frac{5}{6}}$

Put $x-1=t$
$\Rightarrow t\rightarrow 0$ as $x\to 1$
$\therefore \lim\limits_{x\to 1}\big(\large\frac{5}{6}\big)\frac{1-\cos(x-1)}{(x-1)^2}$
$\Rightarrow \lim\limits_{t\to 0}\big(\large\frac{5}{6}\big)^{\Large\frac{1-\cos t}{t^2}}$
$\Rightarrow \big(\large\frac{5}{6}\big)^{\lim\limits_{t\to 0}\Large\frac{2\sin^2t/2}{t^2}}$
$\Rightarrow \big(\large\frac{5}{6}\big)^{\Large\frac{1}{2}}$
$\Rightarrow \sqrt{\large\frac{5}{6}}$
Hence (d) is the correct answer.