If the function

Question:

If the function $f(x)=\frac{\cos (\sin x)-\cos x}{x^{4}}$ is continuous at each point in its domain and $\mathrm{f}(0)=\frac{1}{\mathrm{k}}$, then $\mathrm{k}$ is

Solution:

$\lim _{x \rightarrow 0} \frac{\cos (\sin x)-\cos x}{x^{4}}=f(0)$

$\Rightarrow \lim _{x \rightarrow 0} \frac{2 \sin \left(\frac{\sin x+x}{2}\right) \sin \left(\frac{x-\sin x}{2}\right)}{x^{4}}=\frac{1}{K}$

$\Rightarrow \lim _{x \rightarrow 0} 2\left(\frac{\sin x+x}{2 x}\right)\left(\frac{x-\sin x}{2 x^{3}}\right)=\frac{1}{K}$

$\Rightarrow 2 \times \frac{(1+1)}{2} \times \frac{1}{2} \times \frac{1}{6}=\frac{1}{\mathrm{~K}}$

$\Rightarrow \mathrm{K}=6$

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