Q56 / 90 · JEE Main 2012 (07 May Online) · Mathematics

$\lim _{x \rightarrow 0}\left(\frac{x-\sin x}{x}\right) \sin \left(\frac{1}{x}\right)$

  • A
    equals 1
  • Correct answer
    B
    equals 0
  • C
    does not exist
  • D
    equals $-1$

Solution

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

Hence, the correct answer is option: 'B'.

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