Mathematics - Limits Question with Solution | TestHub

As $x$ approaches $\frac{\pi }{2}$, $\sin x$ approaches 1 and $\cos x$ approaches 0. Therefore, $[\sin x]=0$ and $[\cos x]=0$.

$$\underset{x\to \frac{\pi }{2}}{\lim }\left[\frac{[\sin x]-[\cos x]+1}{3}\right]=\left[\frac{0-0+1}{3}\right]=\left[\frac{1}{3}\right]=0$$

The greatest integer function (GIF) concept is used.