If f:R to R is a differentiable function and | vedclass

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(C) Let $L = \lim_{x 	o 2} \frac{1}{x - 2} \int_{6}^{f(x)} 2t \, dt$. Since $f(2) = 6$,the integral becomes $\int_{6}^{6} 2t \, dt = 0$,leading to a

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$12f'(2)$

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(C) Let $L = \lim_{x 	o 2} \frac{1}{x - 2} \int_{6}^{f(x)} 2t \, dt$. Since $f(2) = 6$,the integral becomes $\int_{6}^{6} 2t \, dt = 0$,leading to a $\frac{0}{0}$ indeterminate form. Applying $L'	ext{H\^opital's Rule}$,we differentiate the numerator and denominator with respect to $x$: Numerator: $\frac{d}{dx} \int_{6}^{f(x)} 2t \, dt = 2f(x) \cdot f'(x)$ (by Leibniz Integral Rule). Denominator: $\frac{d}{dx} (x - 2) = 1$. Thus,$L = \lim_{x 	o 2} \frac{2f(x)f'(x)}{1} = 2f(2)f'(2)$. Given $f(2) = 6$,we get $L = 2(6)f'(2) = 12f'(2)$.

If $f:R \to R$ is a differentiable function and $f(2) = 6$,then $\lim_{x \to 2} \int_{6}^{f(x)} \frac{2t \, dt}{x - 2}$ is

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