Question:
If $f: \mathrm{R} \rightarrow \mathrm{R}$ is a differentiable function and $f(2)=6$, then $\lim _{x \rightarrow 2} \int_{6}^{f(\mathrm{x})} \frac{2 t d t}{(x-2)}$ is:
Correct Option: , 4
Solution:
Using L' Hospital rule and Leibnitz theorem, we get
$\lim _{x \rightarrow 2} \frac{\int_{0}^{f(x)} 2 t d t}{(x-2)}=\lim _{x \rightarrow 2} \frac{2 f(x) f^{\prime}(x)-0}{1}$
Putting $x=2,2 f(2) f^{\prime}(2)=12 f^{\prime}(2) \quad[\because f(2)=6]$
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