A) e
B) \[{{e}^{2}}\]
C) \[\sqrt{e}\]
D) \[1/\sqrt{e}\]
Correct Answer: B
Solution :
\[\underset{x\to 0}{\mathop{\lim }}\,\tan {{\left( \frac{\pi }{4}+x \right)}^{1/x}}=\underset{x\to 0}{\mathop{\lim }}\,{{\left\{ \frac{1+\tan x}{1-\tan x} \right\}}^{1/x}}\] \[=\underset{x\to 0}{\mathop{\lim }}\,{{\left\{ 1+\frac{2\tan x}{1-\tan x} \right\}}^{1/x}}\] \[={{e}^{\underset{x\to 0}{\mathop{\lim }}\,\frac{2\tan x}{1-\tan x}.\frac{1}{x}}}\] \[={{e}^{\underset{x\to 0}{\mathop{\lim }}\,\frac{2}{1-\tan x}.\frac{\tan x}{x}}}\] \[={{e}^{\frac{2}{1-0}\times 1}}={{e}^{2}}\]
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