## Problem Statement

Calculate the limit of the function:

$$
\lim _{x \rightarrow 1}(2-x)^{\sin \left(\frac{\pi x}{2}\right) / \ln (2-x)}
$$
## Solution

Calculate the limit of the function:

$$
\begin{aligned}
& \lim _{x \rightarrow 1}(2-x)^{\sin \left(\frac{\pi x}{2}\right) / \ln (2-x)}= \\
& =\lim _{x \rightarrow 1}\left(e^{\ln (2-x)}\right)^{\sin \left(\frac{\pi x}{2}\right) / \ln (2-x)}= \\
& =\lim _{x \rightarrow 1} e^{\ln (2-x) \cdot \sin \left(\frac{\pi x}{2}\right) / \ln (2-x)}= \\
& =\lim _{x \rightarrow 1} e^{\sin \left(\frac{\pi x}{2}\right)}=e^{\sin \left(\frac{\pi \cdot 1}{2}\right)}= \\
& =e^{\sin \frac{\pi}{2}}=e^{1}=e
\end{aligned}
$$

Problem Kuznetsov Limits 19-25