Question

$\underset{x\to 0}{lim}{\left[\frac{1+\tan x}{1+\sin x}\right]}^{\csc x}$ is equal to

• A. $e$
• B. $\frac{1}{e}$
• C. $1$
• D. None of these

Answer 1

Answer: (C)

$1$

$\text{The given limit is}$

$\underset{x\to 0}{lim}\frac{(1+\tan x{)}^{cosecx}}{(1+\sin x{)}^{cosecx}}$

$=\underset{x\to 0}{lim}\left[{\left\{\right(1+\tan x{)}^{cotx}\}}^{\sec x}\frac{1}{(1+\sin x{)}^{cosecx}}\right]$

$=e^{\sec 0}\frac{1}{e}=e\frac{1}{e}=1$