A) 0
B) 1
C) \[f'(1)\]
D) \[\infty \]
Correct Answer: C
Solution :
\[\underset{x\to 1}{\mathop{\lim }}\,\int_{1}^{f(x)}{\frac{t\,dt}{(x-1)}}=\underset{x\to 1}{\mathop{\lim }}\,=\frac{1}{x-1}\left[ \left( \frac{{{t}^{2}}}{2} \right) \right]_{1}^{f(x)}\] \[=\frac{1}{2}\underset{x\to 1}{\mathop{\lim }}\,\frac{1}{x-1}[{{\{f(x)\}}^{2}}-1]\] \[=\frac{1}{2}\underset{x\to 1}{\mathop{\lim }}\,\frac{[2f(x)f'(x)]}{1}\] [by L' Hospital rule]
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