Prove the following
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:

1. (1) $24 f^{\prime}(2)$

2. (2) $2 f^{\prime}(2)$

3. (3) 0

4. (4) $12 f^{\prime}(2)$

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 $x=2,2 f(2) f^{\prime}(2)=12 f^{\prime}(2) \quad[\because f(2)=6]$