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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} x\log (\sin x) = $
  • A
    $-1$
  • ✓
    ${\log _e}1$
  • C
    $1$
  • D
    None of these
✓

Answer

Correct option: B.
${\log _e}1$
b
(b)$\mathop {\lim }\limits_{x \to 0} x\log \sin x = \mathop {\lim }\limits_{x \to 0} \,\log \,{(\sin x)^x} = \log \,[\mathop {\lim }\limits_{x \to 0} \,\,{(\sin x)^x}]$

$ = \log \,\left[ {\mathop {\lim }\limits_{x \to 0} \,{{(1 + \sin x - 1)}^{\frac{{x(\sin x - 1)}}{{\sin x - 1}}}}} \right]$

$ = {\log _e}[{e^{\mathop {\lim }\limits_{x \to 0} \,x(\sin x - 1)}}] = {\log _e}1.$

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