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\lim_{x \to 0 }\frac{\int_{0}^{x^{2}}\left ( \sin \sqrt{t} \right )dt}{x^{3}}is equal to:
Option: 1 0
Option: 2 \frac{3}{2}
Option: 3 \frac{1}{15}
Option: 4 \frac{2}{3}

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\lim _{x \rightarrow 0^{+}} \frac{\int_{0}^{x^{2}} \sin \sqrt{t} d t}{x^{3}}=\lim _{x \rightarrow 0^{+}} \frac{(\sin |x|) 2 x}{3 x^{2}}=\lim _{x \rightarrow 0^{+}}\left(\frac{\sin x}{x}\right) \times \frac{2}{3}=\frac{2}{3}

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Suraj Bhandari

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