If f: R rightarrow R is defined by f(x) = beg | vedclass

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(B) Given that $f(x) = \begin{cases} \frac{\cos 3x - \cos x}{x^2}, & 	ext{for } x 
eq 0 \ \lambda, & 	ext{for } x = 0 \end{cases}$.Since $f(x)$ is

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$-4$

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(B) Given that $f(x) = \begin{cases} \frac{\cos 3x - \cos x}{x^2}, & 	ext{for } x 
eq 0 \ \lambda, & 	ext{for } x = 0 \end{cases}$.Since $f(x)$ is continuous at $x = 0$,we must have $\lim_{x ightarrow 0} f(x) = f(0) = \lambda$.Evaluating the limit using the formula $\cos C - \cos D = -2 \sin \frac{C+D}{2} \sin \frac{C-D}{2}$:$\lim_{x ightarrow 0} \frac{\cos 3x - \cos x}{x^2} = \lim_{x ightarrow 0} \frac{-2 \sin \frac{3x+x}{2} \sin \frac{3x-x}{2}}{x^2} = \lim_{x ightarrow 0} \frac{-2 \sin 2x \sin x}{x^2}$.Using the standard limit $\lim_{	heta ightarrow 0} \frac{\sin 	heta}{	heta} = 1$:$\lim_{x ightarrow 0} -2 \left( \frac{\sin 2x}{x} ight) \left( \frac{\sin x}{x} ight) = -2 	imes 2 	imes 1 = -4$.Therefore,$\lambda = -4$.

If $f: R \rightarrow R$ is defined by $f(x) = \begin{cases} \frac{\cos 3x - \cos x}{x^2}, & \text{for } x \neq 0 \\ \lambda, & \text{for } x = 0 \end{cases}$ and if $f$ is continuous at $x = 0$,then $\lambda$ is equal to

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