If f:RR is a differentiable function and f(2)=6, then _x2_6^f(x)2t,dtx-2 is:

Question

If $f:\mathbb{R}	o\mathbb{R}$ is a differentiable function and $f(2)=6$, then $\displaystyle\lim_{x	o2}\frac{\displaystyle\int_6^{f(x)}2t\,dt}{x-2}$ is:

Accepted Answer

Using L'Hopital's rule and Leibnitz's theorem (numerator and denominator both $\to0$ as $x\to2$):
$$\lim_{x\to2}\frac{\int_6^{f(x)}2t\,dt}{x-2}=\lim_{x\to2}\frac{2f(x)f'(x)}{1}=2f(2)f'(2)=2\cdot6\cdot f'(2)=12f'(2)$$

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