Evaluate $\underset{x→1}{\lim}\frac{\cos\frac{πx}{2}}{1-\sqrt{x}}$.

$π$

Let x – 1 = t

$\underset{t→0}{\lim}\frac{\cos(\frac{π}{2}t+\frac{π}{2})}{1-\sqrt{1+t}}=\underset{t→0}{\lim}\frac{-\sin(\frac{π}{2}t)}{-t}(1+\sqrt{1+t})=\underset{t→0}{2\lim}\frac{π}{2}\sin\frac{(\frac{π}{2}t)}{\frac{π}{2}t}=π[\underset{h→0}{\lim}\frac{\sin h}{h}=1]$