If f:R R is a differentiable function and f ( 2 ) = 6, then _ x 2… - Vidyadip

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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