Question

$\underset{x\to 0}{lim}\frac{\sin \left[\cos x\right]}{1+\left[\cos x\right]}$ is:
(Note : $[.]$ denotes the greatest integer function)

• A. 1
• B. 0
• C. does not exist
• D. 2

Answer 1

The correct option is B 0

$\underset{x\to 0}{lim}\frac{\sin \left[\cos x\right]}{1+\left[\cos x\right]}$ where [.] is a greatest integer function

$asx\to 0^{-}0<\cos x<1$

$asx\to 0^{+}0<\cos x<1$
in both cases $\left[\cos x\right]=0$

L.H.L = $\underset{x\to 0^{-}}{lim}\frac{\sin \left[\cos x\right]}{1+\left[\cos x\right]}=0$

= $\underset{x\to 0^{-}}{lim}\frac{\sin 0}{1+0}=0$

R.H.L = $\underset{x\to 0^{+}}{lim}\frac{\sin \left[\cos x\right]}{1+\left[\cos x\right]}=0$

= $\underset{x\to 0^{+}}{lim}\frac{\sin 0}{1+0}=0$

$\therefore \underset{x\to 0}{lim}\frac{\sin \left[\cos x\right]}{1+\left[\cos x\right]}=0$ (As, LHL = RHL.)