If \( f(x)=\lim _{n \rightarrow \infty} \frac{(1-\cos (1-\cos x))(x+1)^{n}+\lambda \sin \left(n-...

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If \( f(x)=\lim _{n \rightarrow \infty} \frac{(1-\cos (1-\cos x))(x+1)^{n}+\lambda \sin \left(n-\sqrt{n^{2}-n}\right) x}{x^{4}(x+1)^{n}+x} \), \( x \neq 0 \) is continuous at \( x=0 \), then find the value of \( \lambda \)
(a) \( 1 / 2 \)
(b) \( -1 / 4 \)
(c) \( 1 / 4 \)
(d) \( 1 / 3 \)
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