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

CBSE BoardEnglish MediumSTD 11 ScienceMathsSTD 11 - 12. limits1 Mark
MCQ
If $f:R \to R$ is a differentiable function and $f\left( 2 \right) = 6$, then $\mathop {\lim }\limits_{x \to 2} \int\limits_6^{f\left( x \right)} {\frac{{2\,tdt}}{{\left( {x - 2} \right)}}} $ is
  • A
    $0$
  • B
    $2f'\left( 2 \right)$
  • ✓
    $12f'\left( 2 \right)$
  • D
    $24f'\left( 2 \right)$
✓

Answer

Correct option: C.
$12f'\left( 2 \right)$
c
 $\mathop {\lim }\limits_{x \to 2} \int\limits_6^{f\left( x \right)} {\frac{{2tdt}}{{\left( {x - 2} \right)}}} dx$        {given that $\,f\left( 2 \right) = 6$}

$\frac{0}{0}$ from, so we use $L-$ Hopital Rule

$ = \mathop {\lim }\limits_{x \to 2} \frac{{f'\left( x \right).2f\left( x \right)}}{1}$

$ = f'\left( 2 \right).2f\left( 2 \right)$

$ = 12f'\left( 2 \right)$

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