# If  $f:R\rightarrow R$  is a differentiable function and $F(2)=6$   , then $\lim_{x\rightarrow 2}\int_{6}^{f(x)}\frac{2t\:dt}{(x-2)}$   is : Option 1) $24f^{'}(2)$ Option 2) $2f^{'}(2)$ Option 3) $0$ Option 4) $12f^{'}(2)$

P Plabita

$\\f(2)=6\\\\\:\lim_{x\rightarrow 2}\int_{6}^{f(x)}\frac{2t.dt}{(x-2)}\\\\\:=\lim_{x\rightarrow 2}\frac{t^{2}}{x-2}\; |_{6}^{f(x)}=\lim_{x\rightarrow 2}\frac{\left ( f(x) \right )^{2}-(6)^{2}}{(x-2)}\\\\\:$

$\\=\lim_{x\rightarrow 2}\frac{2f(x)f^{'}(x)}{1}\\\\\:=2(x).f^{'}(x)\\\\\:2 \times 6\times f^{'}(2)=12f^{'}(2)$

Option 1)

$24f^{'}(2)$

Option 2)

$2f^{'}(2)$

Option 3)

$0$

Option 4)

$12f^{'}(2)$

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