Question

$\underset{x\to 0}{lim}\left[\frac{sin\left(sgn\right(x\left)\right)}{\left(sgn\right(x\left)\right)}\right]$, where $[.]$ denotes the greatest integer function, is equal to

• A. 0
• B. 1
• C. -1
• D. Does not exist

Answer 1

The correct option is A 0

$RHL=\underset{x\to 0^{+}}{lim}\left[\frac{\sin \left(sgnx\right)}{sgn\left(x\right)}\right]=\underset{x\to 0^{+}}{lim}\left[\frac{\sin 1}{1}\right]=0$
$LHL=\underset{x\to 0^{-}}{lim}\left[\frac{\sin \left(sgnx\right)}{sgn\left(x\right)}\right]=\underset{x\to 0^{-}}{lim}\left[\frac{\sin (-1)}{-1}\right]=\underset{x\to 0^{-}}{lim}\left[sin1\right]=0$

Hence, the limit is 0.