## Problem Statement

Calculate the limit of the function:

$\lim _{x \rightarrow \frac{\pi}{2}} \frac{2+\cos x \cdot \sin \frac{2}{2 x-\pi}}{3+2 x \sin x}$
## Solution

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

$\cos x \cdot \sin \frac{2}{2 x-\pi} \rightarrow 0$ as $x \rightarrow \frac{\pi}{2}$

Then:

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