1. Tardigrade
  2. Question
  3. Mathematics
  4. displaystyle limx →0 ((x- sin x/x)) sin ((1/x))

Question Error Report

Thank you for reporting, we will resolve it shortly

Back to Question

Q. $\displaystyle\lim_{x \to0} \left(\frac{x-\sin x}{x}\right) \sin \left(\frac{1}{x}\right) $

Limits and Derivatives

Solution:

Consider $\displaystyle\lim_{x \to0} \left(\frac{x-\sin x}{x}\right) \sin \left(\frac{1}{x}\right) $
$ = \displaystyle\lim _{x \to 0} \left[\frac{x\left(1- \frac{\sin x}{x}\right)}{x}\right] \times\displaystyle\lim _{x \to 0} \sin\left(\frac{1}{x}\right) $
$ = \displaystyle\lim _{x \to 0} \left[1- \frac{\sin x}{x}\right] \times\displaystyle\lim _{x \to 0} \sin\left(\frac{1}{x}\right)$
$ = \left[1- \displaystyle\lim _{x \to 0} \frac{\sin x}{x}\right] \times \displaystyle\lim _{x \to 0} \sin\left(\frac{1}{x}\right) $
$ = 0 \times\displaystyle\lim _{x \to 0} \sin \left(\frac{1}{x}\right)= 0 $