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If $f(x)=\left\{\begin{aligned} \frac{1-\cos x}{x^{2}}, & x \neq 0 \\ k &, x=0 \end{aligned}\right.$ is continuous at $x=0$, find $k$.


Given: $f(x)=\left\{\begin{array}{l}\frac{1-\cos x}{x^{2}}, x \neq 0 \\ k, x=0\end{array}\right.$

If $f(x)$ is continuous at $x=0$, then

$\lim _{x \rightarrow 0} f(x)=f(0)$

$\Rightarrow \lim _{x \rightarrow 0}\left(\frac{1-\cos x}{x^{2}}\right)=k$

$\Rightarrow \lim _{x \rightarrow 0}\left(\frac{2\left[\sin \left(\frac{x}{2}\right)\right]^{2}}{4\left(\frac{x}{2}\right)^{2}}\right)=k$

$\Rightarrow \frac{1}{2} \lim _{x \rightarrow 0}\left(\frac{\left[\sin \left(\frac{x}{2}\right)\right]^{2}}{\left(\frac{x}{2}\right)^{2}}\right)=k$

$\Rightarrow 1 \times \frac{1}{2}=k$

$\Rightarrow k=\frac{1}{2}$

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