Question

$\underset{x\to 0}{\lim }{\left(\tan \left(\frac{\pi }{4}+x\right)\right)}^{1/x}$ is equal to

• A. 2
• B. e
• C. 1
• D. $e^2$

Answer 1

Answer: (D)

$\underset{x\to 0}{\lim }{\left\{\tan \left(\frac{\pi }{4}+x\right)\right\}}^{1/x}$
$=e^{\underset{x\to 0}{\lim }\frac{1}{x}\left\{\tan \left(\frac{\pi }{4}+x\right)-1\right\}}$
$=e^{\underset{x\to 0}{\lim }\left(\frac{1+\tan x-1+\tan x}{x(1-\tan x)}\right)}$
$=e^{\underset{x\to 0}{\lim }\frac{2\tan x}{x(1-\tan x)}}$
$=e^2$