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Question
\[\lim_{x \to \frac{\pi}{4}} \frac{\sqrt{\cos x} - \sqrt{\sin x}}{x - \frac{\pi}{4}}\]
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Solution
\[\lim_{x \to \frac{\pi}{4}} \frac{\sqrt{\cos x} - \sqrt{\sin x}}{x - \frac{\pi}{4}}\]
\[\text{ Rationalising the numerator }: \]
\[ = \lim_{x \to \frac{\pi}{4}} \left( \frac{\sqrt{\cos x} - \sqrt{\sin x}}{x - \frac{\pi}{4}} \right) \left( \frac{\sqrt{\cos x} + \sqrt{\sin x}}{\sqrt{\cos x} + \sqrt{\sin x}} \right)\]
\[ = \lim_{x \to \frac{\pi}{4}} \frac{\cos x - \sin x}{\left( x - \frac{\pi}{4} \right) \left( \sqrt{\cos x} + \sqrt{\sin x} \right)}\]
\[ = \lim_{h \to 0} \frac{\cos \left( \frac{\pi}{4} + h \right) - \sin \left( \frac{\pi}{4} + h \right)}{\left( \frac{\pi}{4} + h - \frac{\pi}{4} \right) \left( \sqrt{\cos \left( \frac{\pi}{4} + h \right)} + \sqrt{\sin \left( \frac{\pi}{4} + h \right)} \right)}\]
\[ = \lim_{h \to 0} \frac{\cos \frac{\pi}{4} \cos h - \sin \frac{\pi}{4} \sin h - \sin \frac{\pi}{4} \cos h - \cos \frac{\pi}{4} \sin h}{h\left( \sqrt{\cos \left( \frac{\pi}{4} + h \right)} + \sqrt{\sin \left( \frac{\pi}{4} + h \right)} \right)}\]
\[ = \lim_{h \to 0} \frac{- \sqrt{2} \sin h}{h\left( \sqrt{\cos \left( \frac{\pi}{4} + h \right)} + \sqrt{\sin \left( \frac{\pi}{4} + h \right)} \right)}\]
\[ \Rightarrow \frac{- \sqrt{2} \times 1}{2 \left( \frac{1}{2} \right)^\frac{1}{4}}\]
\[ = \frac{- 1}{2^\frac{1}{4}} = - 2^{- \frac{1}{4}}\]
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