If f : R → R be a differentiable function having f (2) = 6, f'(2) = (1/48). Then limx→2 [∫6f (x) 4t3 dt] / [x - 2] =

1) 18

2) 12

3) 36

4) 24

Solution: (1) 18

\(\begin{array}{l}\begin{array}{l} \lim _{x \rightarrow 2} \frac{\int_{6}^{f(x)} 4 t^{3} d t}{x-2} \quad\left(\frac{0}{0} \text { form }\right)\\ \Rightarrow \lim _{x \rightarrow 2} \frac{4(f(x))^{3} f^{\prime}(x)}{1}\\ \Rightarrow 4(f(x))^{3} f^{\prime}(2) \\\Rightarrow 4(6)^{3}\left(\frac{1}{48}\right)\\=18\\ \text { Alter }\\ \lim _{x \rightarrow 2} \frac{\left[4 \frac{t^{4}}{4}\right]_{6}^{f(x)}}{x-2} \\\Rightarrow \lim _{x \rightarrow 2} \frac{(f(x))^{4}-6^{4}}{x-2}\\ \Rightarrow \lim _{x \rightarrow 2} \frac{4(f(x))^{3} f^{\prime}(x)-0}{1-0} \\ \Rightarrow 4 f(2)^{3} f^{\prime}(2)\\ \Rightarrow 4(6)^{3}\left(\frac{1}{48}\right)\\=18 \end{array}\end{array} \)

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