Question

The value of $\underset{x\to 0}{lim}\left[\frac{\sin |x|}{|x|}\right]$ is
(where [.] is greatest integer function)

• A. Right hand limit (RHL) is $1$.
• B. Left hand limits (LHL) is $-1$.
• C. Both LHL and RHL is 0.
• D. Both LHL and RHL is $1$.

Answer 1

The correct option is C Both LHL and RHL is 0.

LHL at $x=0$:
$\underset{h\to 0}{lim}\left[\frac{\sin |0-h|}{|0-h|}\right]=\underset{h\to 0}{lim}\left[\frac{\sin |-h|}{|-h|}\right]=\underset{h\to 0}{lim}\left[\frac{\sin h}{h}\right]$
We know that,
$\frac{\sin x}{x}<1$
So LHL
$=0$

Finding RHL
$\underset{h\to 0}{lim}\left[\frac{\sin |0+h|}{|0+h|}\right]=\underset{h\to 0}{lim}\left[\frac{\sin |h|}{|h|}\right]=\underset{h\to 0}{lim}\left[\frac{\sin h}{h}\right]$
We know that,
$\frac{\sin x}{x}<1$
So RHL
$=0$

Hence both LHL and RHL is $0$.