Question

The value of $\underset{x\to \pi /2}{\lim }(\sec x{)}^{\cot x}$ is

• A. $1$
• B. $e$
• C. $e^2$
• D. $1/e$

Answer 1

Answer: (A)

$y=\underset{x\to \pi /2}{\lim }(\sec x{)}^{\cot x}({\infty }^0$ form)
$\therefore \log y=\underset{x\to \pi /2}{\lim }\cot x\log \sec x$
$=\underset{x\to \pi /2}{\lim }\frac{\log \sec x}{\tan x}(\frac{\infty }{\infty }$ form $)$
$=\underset{x\to \pi /2}{\lim }\frac{1}{\sec x}\times \frac{\sec x\tan x}{{\sec }^{2}x}$ (Using L'Hospital's Rule)
$=\underset{x\to \pi /2}{\lim }\frac{\tan x}{{\sec }^{2}x}=\underset{x\to \pi /2}{\lim }\sin x\cos x=0$
$\therefore y=e^0=1$