lim_{x rightarrow 0} frac{sin(pi sin^2 x)}{x^ | vedclass

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Question

(D) We use the limit formula $\lim_{	heta ightarrow 0} \frac{\sin 	heta}{	heta} = 1$. Given expression: $\lim_{x ightarrow 0} \frac{\sin(\pi \s

Vedclass · Accepted Answer

$\pi$

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(D) We use the limit formula $\lim_{	heta ightarrow 0} \frac{\sin 	heta}{	heta} = 1$. Given expression: $\lim_{x ightarrow 0} \frac{\sin(\pi \sin^2 x)}{x^2}$. Multiply and divide by $\pi \sin^2 x$: $= \lim_{x ightarrow 0} \left[ \frac{\sin(\pi \sin^2 x)}{\pi \sin^2 x} 	imes \frac{\pi \sin^2 x}{x^2} ight]$. Since $\pi \sin^2 x ightarrow 0$ as $x ightarrow 0$, the first part approaches $1$. $= 1 	imes \pi 	imes \lim_{x ightarrow 0} \left( \frac{\sin x}{x} ight)^2$. $= \pi 	imes (1)^2 = \pi$.

$\lim_{x \rightarrow 0} \frac{\sin(\pi \sin^2 x)}{x^2} = $

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