A) \[0\]
B) \[1\]
C) \[3\]
D) None of these
Correct Answer: D
Solution :
\[\underset{x\to 0}{\mathop{\lim }}\,\,\frac{\int_{0}^{x}{\sin \,({{t}^{2}})dt}}{{{x}^{3}}}\] \[\left( from\,\frac{0}{0} \right)\] \[=\underset{x\to 0}{\mathop{\lim }}\,\,\frac{\sin ({{x}^{2}})}{3{{x}^{2}}}\] (L? Hospital?s rule) \[=\frac{1}{3}\,\underset{x\to 0}{\mathop{\lim }}\,\,\,\,\frac{\sin \,({{x}^{2}})}{{{x}^{2}}}\] \[=\frac{1}{3}\times 1=\frac{1}{3}\] \[\left( \because \,\,\underset{x\to 0}{\mathop{\lim }}\,\,\,\frac{\sin x}{x}=1 \right)\]
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