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12. limits — MATHS STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceMATHS12. limits1 Mark
MCQ
$\mathop {\lim }\limits_{x \to 0} \frac{{{e^{\tan x}} - {e^x}}}{{\tan x - x}} = $
  • ✓
    $1$
  • B
    $e$
  • C
    ${e^{ - 1}}$
  • D
    $0$
✓

Answer

Correct option: A.
$1$
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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