# Lim X → π 4 Cos X − Sin X ( π 4 − X ) ( Cos X + Sin X ) - Mathematics

$\lim_{x \to \frac{\pi}{4}} \frac{\cos x - \sin x}{\left( \frac{\pi}{4} - x \right) \left( \cos x + \sin x \right)}$

#### 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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#### APPEARS IN

RD Sharma Class 11 Mathematics Textbook
Chapter 29 Limits
Exercise 29.8 | Q 37 | Page 63