\[ \begin{split} &\lim_{x \to \dfrac{\pi }{2}}\frac{1-\sin x+\cos x}{\sin 2x-\cos x} =\left[\frac{0}{0}\right]^H=\\[6pt] &=\lim_{x \to \dfrac{\pi }{2}} \frac{-\cos x-\sin x}{2\cos 2x+\sin x}=\\[6pt] &=\frac{-1}{-1}=1 \end{split} \]