Question

The value of $\underset{x\to 0}{\lim }{\left(\cos x\right)}^{{\cot }^{2}x}$ 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 $\underset{x\to 0}{\lim }[1+f(x){]}^{m/f(x)}$ where $f(x)\to 0$ at $x=0$
then its solution is $e^{\underset{x\to 0}{\lim }mf(x)}$
$\underset{x\to 0}{\lim }(\cos x{)}^{{\cot }^{2}x}=\underset{x\to 0}{\lim }{\left(1-2{\sin }^2\frac{x}{2}\right)}^{{\cot }^{2}x}$
$=e^{-\underset{x\to 0}{\lim }2{\sin }^2\frac{x}{2}{\cot }^{2}x}$
$=e-\underset{x\to 0}{\lim }2{\sin }^2\frac{x}{2}\times \frac{{\cos }^{2}x}{4{\sin }^2\frac{x}{2}{\cos }^2\frac{x}{2}}$
$-\underset{x\to 0}{\lim }\frac{{\cos }^{2}x}{2{\cos }^2\frac{x}{2}}$
$=e^{-\frac{1}{2}}$