Question

Let $f(x)=\frac{1}{3} x \sin x-(1-\cos x)$. The smallest positive interger $k$ such that $\lim\limits _{x \rightarrow 0} \frac{f(x)}{x} \neq 0$

• A. 4
• B. 3
• C. 2
• D. 1

Answer 1

Answer: (C)

$\lim _{x \rightarrow 0} \frac{f(x)}{x^{k}}=\lim _{x \rightarrow 0} \frac{\frac{x \sin x}{3}-1+\cos x}{x^{k}}\left[\frac{0}{0}\right.$ form $]$
$=\lim _{x \rightarrow 0} \frac{\frac{\sin x}{3}+\frac{x \cos x}{3}-\sin x}{k x^{k-1}}\left[\frac{0}{0}\right.$ form $]$
$=\lim _{x \rightarrow 0} \frac{\frac{\cos x}{3}+\frac{\cos x}{3}-\frac{x \sin x}{3}-\cos x}{k(k-1) x^{k-2}} \neq 0$ if $k=2$