Question

Find: \( \lim_{x \to 0} \frac{| \sin x |}{x} \)

• A. 1
• B. -1
• C. Does not exist
• D. None of these

Hint:

For limits involving absolute value functions, check the one-sided limits separately. If they do not match, the two-sided limit does not exist.

Answer 1

The Correct Option is C

Step 1: Consider the given limit \( \lim_{x \to 0} \frac{| \sin x |}{x} \). 
For \( x \to 0^+ \) (approaching 0 from the right), \( \sin x \) is positive, so \( | \sin x | = \sin x \). 
The limit becomes: \[ \lim_{x \to 0^+} \frac{\sin x}{x} = 1. \] For \( x \to 0^- \) (approaching 0 from the left), \( \sin x \) is negative, so \( | \sin x | = -\sin x \). The limit becomes: \[ \lim_{x \to 0^-} \frac{-\sin x}{x} = -1. \] 
Since the limit from the right is 1 and the limit from the left is -1, the two one-sided limits are not equal. 
Hence, the limit does not exist.