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STD 11 - 12. limits — JEE STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceJEESTD 11 - 12. limits4 Marks
Question
$\mathop {\lim }\limits_{x \to 0} \frac{{{e^{\tan x}} - {e^x}}}{{\tan x - x}} = $
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Answer

a
(a) $\mathop {\lim }\limits_{x \to 0} \,\,\frac{{{e^{\tan x}} - {e^x}}}{{\tan x - x}} = \mathop {\lim }\limits_{x \to 0} \,\,\frac{{{e^x}[{e^{\tan x - x}} - 1]}}{{\tan x - x}}$

$ = \mathop {\lim }\limits_{x \to 0} \,{e^x}\,.\mathop {\lim }\limits_{x \to 0} \,\frac{{{e^{\tan x - x}} - 1}}{{\tan x - x}} = {e^0} \times 1 = 1$.

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