If f(x) = begin{cases} frac{1 - cos x}{x}, & | vedclass

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(A) For a function $f(x)$ to be continuous at $x = 0$,the limit of $f(x)$ as $x 	o 0$ must equal $f(0)$.Given $f(0) = k$.We need to evaluate $\lim_{x

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(A) For a function $f(x)$ to be continuous at $x = 0$,the limit of $f(x)$ as $x 	o 0$ must equal $f(0)$.Given $f(0) = k$.We need to evaluate $\lim_{x 	o 0} f(x) = \lim_{x 	o 0} \frac{1 - \cos x}{x}$.Using the trigonometric identity $1 - \cos x = 2 \sin^2(\frac{x}{2})$,we get:$\lim_{x 	o 0} \frac{2 \sin^2(\frac{x}{2})}{x} = \lim_{x 	o 0} \left( \frac{\sin(\frac{x}{2})}{\frac{x}{2}} ight)^2 \cdot \frac{x}{2} = (1)^2 \cdot 0 = 0$.Since the limit is $0$,for the function to be continuous,$k$ must be $0$.

If $f(x) = \begin{cases} \frac{1 - \cos x}{x}, & x \ne 0 \\ k, & x = 0 \end{cases}$ is continuous at $x = 0$,then $k = $

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