Question

The value of $\underset{x\to \cos 3}{lim}\left[{\cos }^{-1}x\right]$ where [ . ] denotes greatest integer function is

• A. $2$
• B. $3$
• C. $-3$
• D. Limit does not exist

Answer 1

The correct option is D Limit does not exist

$\underset{x\to cos3}{lim}\left[co{s}^{-1}x\right]$

$LHL=\underset{x\to cos{3}^{-}}{lim}\left[co{s}^{-1}x\right]=\underset{h\to 0}{lim}\left[{\cos }^{-1}\right(\cos 3-h\left)\right]=\underset{h\to 0}{lim}[3-h]=2$

$RHL=\underset{x\to cos{3}^{+}}{lim}\left[co{s}^{-1}x\right]=\underset{h\to 0}{lim}\left[{\cos }^{-1}\right(\cos 3+h\left)\right]=\underset{h\to 0}{lim}[3+h]=3$

$LHL\ne RHL$
$\therefore$ Limits does not exist