Question

Evaluate $li{m}_{x\to 0}\frac{si{n}^{-1}\left[cosx\right]}{1+\left[cosx\right]}$ ([.] denotes the greatest integer function).

• A. $0$
• B. $-1$
• C. $1$
• D. $\infty$

Answer 1

Answer: (A)

$0$

$\underset{x\to 0}{lim}\frac{{\sin }^{-1}(\cos x-1)}{1+\cos x-1}<\underset{x\to 0}{lim}\frac{{\sin }^{-1}\left[\cos x\right]}{1+\left[\cos x\right]}<\underset{x\to 0}{lim}\frac{{\sin }^{-1}(\cos x+1)}{1+\cos x+1}$

$\Rightarrow \underset{x\to 0}{lim}\frac{{\sin }^{-1}0}{1+1-1}<\underset{x\to 0}{lim}\frac{{\sin }^{-1}\left[\cos x\right]}{1+\left[\cos x\right]}<\underset{x\to 0}{lim}\frac{{\sin }^{-1}(\cos x+1)}{1+1+1}$

$\Rightarrow \underset{x\to 0}{lim}\frac{{\sin }^{-1}\left[\cos x\right]}{1+\left[\cos x\right]}=0$
Hence, option 'A' is correct.