## Problem Statement

Calculate the limit of the function:
$\lim _{x \rightarrow \frac{\pi}{2}} \sqrt{3 \sin x+(2 x-\pi) \sin \frac{x}{2 x-\pi}}$
## Solution

Since $\sin \frac{x}{2 x-\pi}$ is bounded, and $(2 x-\pi) \rightarrow 0$, as $x \rightarrow \frac{\pi}{2}$, then

$$
(2 x-\pi) \sin \frac{x}{2 x-\pi} \rightarrow{ }_{, \text{as }}^{2 x} x \rightarrow \frac{\pi}{2}
$$

Then:

$$
\lim _{x \rightarrow \frac{\pi}{2}} \sqrt{3 \sin x+(2 x-\pi) \sin \frac{x}{2 x-\pi}}=\sqrt{3 \sin \frac{\pi}{2}+0}=\sqrt{3 \cdot 1}=\sqrt{3}
$$