群里一极限题 $\lim_{x\to2}(\cos\frac\pi x)/(2-\sqrt{2x})$

\[
\lim_{x \to 2} \frac{\cos\frac\pi x}{2 - \sqrt {2x}} = \lim_{x \to 2} \frac{(2 + \sqrt {2x} )\sin \frac{\pi (x - 2)}{2x}}{2(2 - x)} = \lim_{x \to 2} \left(\frac{ - \pi (2 + \sqrt {2x} )}{4x} \cdot \frac{\sin \frac{\pi (x - 2)}{2x}}{\frac{\pi (x - 2)}{2x}}\right) = \lim_{x \to 2} \frac{ - \pi (2 + \sqrt {2x} )}{4x} = - \frac\pi2
\]