limlimits_{x rightarrow 0} frac{intlimits_{0} | vedclass

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(A) To evaluate the limit $L = \lim_{x \rightarrow 0} \frac{\int_{0}^{x} t \sin(10t) dt}{x}$,we observe that it is of the indeterminate form $\frac{0}

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(A) To evaluate the limit $L = \lim_{x ightarrow 0} \frac{\int_{0}^{x} t \sin(10t) dt}{x}$,we observe that it is of the indeterminate form $\frac{0}{0}$.Applying $L'H\hat{o}pital's$ rule,we differentiate the numerator and the denominator with respect to $x$.Using the $Leibniz$ integral rule for the numerator: $\frac{d}{dx} \int_{0}^{x} t \sin(10t) dt = x \sin(10x)$.The derivative of the denominator $x$ with respect to $x$ is $1$.Thus,the limit becomes $\lim_{x ightarrow 0} \frac{x \sin(10x)}{1}$.As $x ightarrow 0$,$x \sin(10x) ightarrow 0 	imes \sin(0) = 0 	imes 0 = 0$.

$\lim\limits_{x \rightarrow 0} \frac{\int\limits_{0}^{x} t \sin (10 t) d t}{x}$ is equal to

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