Question

The value of $ \lim\limits_{x\to0}\left(\cos\,x\right)^{\cot^2\,x} $ is

• A. $ e^{-1} $
• B. $ e^{−1/2} $
• C. 1
• D. not existing

Answer 1

Answer: (B)

If the limit is of the form of $\lim\limits _{x \rightarrow 0}[1+f(x)]^{m / f(x)}$ where $f(x) \rightarrow 0$ at $x=0$
then its solution is $e^{\lim\limits _{x \rightarrow 0} m f(x)}$
$\lim\limits _{x \rightarrow 0}(\cos x)^{\cot ^{2} x}=\lim\limits _{x \rightarrow 0}\left(1-2 \sin ^{2} \frac{x}{2}\right)^{\cot ^{2} x}$
$=e^{-\lim\limits _{x \rightarrow 0} 2 \sin ^{2} \frac{x}{2} \cot ^{2} x}$
$=e -\lim\limits _{x \rightarrow 0} 2 \sin ^{2} \frac{x}{2} \times \frac{\cos ^{2} x}{4 \sin ^{2} \frac{x}{2} \cos ^{2} \frac{x}{2}}$
$-\lim\limits _{x \rightarrow 0} \frac{\cos ^{2} x}{2 \cos ^{2} \frac{x}{2}}$
$= e^{-\frac{1}{2}} $