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Question
\[\lim_{x \to \frac{\pi}{4}} \frac{\cos x - \sin x}{\left( \frac{\pi}{4} - x \right) \left( \cos x + \sin x \right)}\]
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Solution
\[\lim_{x \to \frac{\pi}{4}} \frac{\cos x - \sin x}{\left( \frac{\pi}{4} - x \right) \left( \cos x + \sin x \right)}\]
\[\text{ Dividing the numerator and the denominator by }\sqrt{2}:\]
\[ \lim_{x \to \frac{\pi}{4}} \frac{\frac{1}{\sqrt{2}} \cos x - \frac{1}{\sqrt{2}} \sin x}{\left( \frac{\pi}{4} - x \right) \frac{\left( \cos x + \sin x \right)}{\sqrt{2}}}\]
\[ = \lim_{x \to \frac{\pi}{4}} \frac{\sqrt{2} \left( \sin \frac{\pi}{4} \cos x - \cos \frac{\pi}{4} \sin x \right)}{\left( \frac{\pi}{4} - x \right) \left( \cos x + \sin x \right)}\]
\[ = \lim_{x \to \frac{\pi}{4}} \frac{\sqrt{2} \left( \sin \left( \frac{\pi}{4} - x \right) \right)}{\left( \frac{\pi}{4} - x \right) \left( \cos x + \sin x \right)}\]
\[ = \lim_{x \to \frac{\pi}{4}} \frac{\sqrt{2}}{\sin x + \cos x} \times \lim_{x \to \frac{\pi}{4}} \frac{\sin \left( \frac{\pi}{4} - x \right)}{\left( \frac{\pi}{4} - x \right)}\]
\[ \Rightarrow \frac{\sqrt{2}}{\frac{1}{\sqrt{2}} + \frac{1}{\sqrt{2}}} \times 1\]
\[ = 1\]
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