Question

$\underset{x\to \pi }{lim}(1-4\tan x{)}^{cotx}=$

• A. $e$
• B. $e^4$
• C. $e^{-1}$
• D. $e^{-4}$

Answer 1

Answer: (D)

$e^{-4}$

Let, $\text{L}=\underset{x\to \pi }{lim}(1-4\tan x{)}^{cotx}$
Put $y=\pi -x\Rightarrow x\to \pi \iff y\to 0$
$\therefore \text{L}=\underset{y\to 0}{lim}(1-4\tan (\pi -y){)}^{cot(\pi -y)}$
$=\underset{y\to 0}{lim}(1+4\tan y{)}^{-coty}[\because \tan (\pi -y)=-\tan y]$
Clearly form of the limit is $1^{\infty }$
$\therefore \text{L}=\underset{y\to 0}{lim}(1+4\tan y{)}^{\frac{1}{4\tan y}\times (-4)}=e^{-4}$