Question

Let \(f(x)=\frac{1}{3} x \sin x-(1-\cos x)\). The smallest positive interger \(k\) such that \(\lim _{x \rightarrow 0} \frac{f(x)}{x^{k}} \neq 0\) is
(A) 4
(B) 3
(C) 2
(D) 1

Answer 1

Ans: (C)
Hint : \(\operatorname{lt}_{x \rightarrow 0} \frac{x \sin x-3(1-\cos x)}{3 x^{k}}=\frac{1}{3} \operatorname{lt}_{x \rightarrow 0}\left(\frac{\sin x / 2}{x / 2}\right) \operatorname{lit}_{x \rightarrow 0}\left(\frac{2 x \cos x / 2-6 \sin x / 2}{2 x^{k-1}}\right)\) \(\mathrm{k}-1=1 \Rightarrow \mathrm{k}=2\)

Answer 2

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Answer 3

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