\[ \begin{split} &\lim_{x \to \dfrac{\pi }{4}}\frac{\sqrt{\sin x}-\sqrt{\cos x}}{\sin x-\cos x} =\\[16pt] &=\lim_{x \to \dfrac{\pi }{4}}\frac{(\sqrt{\sin x}-\sqrt{\cos x})}{(\sqrt{\sin x}-\sqrt{\cos x})(\sqrt{\sin x}+\sqrt{\cos x})}=\\[16pt] &=\lim_{x \to \dfrac{\pi }{4}}\frac{1}{\sqrt{\sin x}+\sqrt{\cos x}}=\\[16pt] &=\frac{1}{\sqrt{\frac{\sqrt{2}}{2}}+\sqrt{\frac{\sqrt{2}}{2}}}=\frac{1}{\sqrt{2\sqrt{2}}} \end{split} \]