Q22 / 50 · Manipal (MET) 2023 · Mathematics
If $F: R \rightarrow R$ is a differentiable function and $f(3)=6$, then $\lim _{x \rightarrow 3} \int_6^{f(x)} \frac{2 t d t}{(x-2)}$ is equal to
- Correct answerA$18 f^{\prime}(3)$
- B$0$
- C$24 f^{\prime}(3)$
- D$3 f^{\prime}(3)$
Solution
Let $I=\lim _{x \rightarrow 3} \int_6^{f(x)} \frac{2 t}{x-2} d t\left[\frac{0}{0}\right.$ form, as $\left.f(3)=6\right]$
On applying the L' Hospital rule, we get
$I=\lim _{x \rightarrow 3} \int_6^{f(x)} \frac{2 f(x) f^{\prime}(x)}{1}$ $(\because f(3)=6)$
So, $I=3 f(3) \cdot f^{\prime}(3)=18 f^{\prime}(3)$
$\therefore \quad \lim _{x \rightarrow 3} \int_6^{f(x)} \frac{2 t d t}{x-2}=18 f^{\prime}(3)$
Hence, the correct answer is option: 'A'.