Advertisement Remove all ads

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

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

Advertisement Remove all ads

Solution

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

  Is there an error in this question or solution?
Advertisement Remove all ads

APPEARS IN

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

Video TutorialsVIEW ALL [1]

Advertisement Remove all ads
Share
Notifications

View all notifications


      Forgot password?
View in app×