Question

$\underset{x\to 0}{\lim }\left(\frac{x-\sin x}{x}\right)\sin \left(\frac{1}{x}\right)$

• A. equals 1
• B. equals 0
• C. does not exist
• D. equals - 1

Answer 1

Answer: (B)

Consider $\underset{x\to 0}{\lim }\left(\frac{x-\sin x}{x}\right)\sin \left(\frac{1}{x}\right)$
$=\underset{x\to 0}{\lim }\left[\frac{x\left(1-\frac{\sin x}{x}\right)}{x}\right]\times \underset{x\to 0}{\lim }\sin \left(\frac{1}{x}\right)$
$=\underset{x\to 0}{\lim }\left[1-\frac{\sin x}{x}\right]\times \underset{x\to 0}{\lim }\sin \left(\frac{1}{x}\right)$
$=\left[1-\underset{x\to 0}{\lim }\frac{\sin x}{x}\right]\times \underset{x\to 0}{\lim }\sin \left(\frac{1}{x}\right)$
$=0\times \underset{x\to 0}{\lim }\sin \left(\frac{1}{x}\right)=0$