The value of mathop {lim }limits_{x to 0} fra | vedclass

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(B) Let $L = \mathop {\lim }\limits_{x 	o 0} \frac{{\int_0^x {\cos {t^2}dt} }}{x}$.Since the limit is of the form $\frac{0}{0}$,we apply $L'	ext{Hos

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$1$

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(B) Let $L = \mathop {\lim }\limits_{x 	o 0} \frac{{\int_0^x {\cos {t^2}dt} }}{x}$.Since the limit is of the form $\frac{0}{0}$,we apply $L'	ext{Hospital's rule}$.Using the $	ext{Leibniz integral rule}$,the derivative of the numerator $\frac{d}{dx} \int_0^x \cos(t^2) dt = \cos(x^2)$.The derivative of the denominator $\frac{d}{dx} (x) = 1$.Therefore,$L = \mathop {\lim }\limits_{x 	o 0} \frac{\cos(x^2)}{1} = \cos(0) = 1$.

The value of $\mathop {\lim }\limits_{x \to 0} \frac{{\int_0^x {\cos {t^2}dt} }}{x}$ is

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