If f(x) = frac{sin(pi cos^2 x)}{3x^2} for x n | vedclass

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(B) For $f(x)$ to be continuous at $x = 0$,we must have $f(0) = \lim_{x 	o 0} f(x)$.Given $f(x) = \frac{\sin(\pi \cos^2 x)}{3x^2}$.Using the identity

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$\frac{\pi}{3}$

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(B) For $f(x)$ to be continuous at $x = 0$,we must have $f(0) = \lim_{x 	o 0} f(x)$.Given $f(x) = \frac{\sin(\pi \cos^2 x)}{3x^2}$.Using the identity $\cos^2 x = 1 - \sin^2 x$,we have:$f(x) = \frac{\sin(\pi(1 - \sin^2 x))}{3x^2} = \frac{\sin(\pi - \pi \sin^2 x)}{3x^2}$.Since $\sin(\pi - 	heta) = \sin 	heta$,we get:$f(x) = \frac{\sin(\pi \sin^2 x)}{3x^2}$.Now,$\lim_{x 	o 0} f(x) = \lim_{x 	o 0} \frac{\sin(\pi \sin^2 x)}{3x^2} = \lim_{x 	o 0} \left( \frac{\sin(\pi \sin^2 x)}{\pi \sin^2 x} 	imes \frac{\pi \sin^2 x}{3x^2} ight)$.Since $\lim_{u 	o 0} \frac{\sin u}{u} = 1$ and $\lim_{x 	o 0} \frac{\sin x}{x} = 1$,we have:$f(0) = 1 	imes \frac{\pi}{3} 	imes (1)^2 = \frac{\pi}{3}$.

If $f(x) = \frac{\sin(\pi \cos^2 x)}{3x^2}$ for $x \neq 0$ is continuous at $x = 0$,then $f(0) = $

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