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