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Lim θ → π / 2 1 − Sin θ ( π / 2 − θ ) Cos θ is Equal to - Mathematics

MCQ

\[\lim_\theta \to \pi/2 \frac{1 - \sin \theta}{\left( \pi/2 - \theta \right) \cos \theta}\] is equal to 

Options

  •  1

  • −1 

  • \[\frac{1}{2}\]

  • \[- \frac{1}{2}\] 

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Solution

\[\frac{1}{2}\] 

\[\lim_\theta \to \frac{\pi}{2} \frac{1 - \sin \theta}{\left( \frac{\pi}{2} - \theta \right)\cos \theta}\]
\[ = \lim_{h \to 0} \frac{1 - \cos h}{\left( \frac{\pi}{2} - \left( \frac{\pi}{2} - h \right) \right) \sin h}\]
\[ = \lim_{h \to 0} \frac{2 \sin^2 \frac{h}{2}}{h \sin h}\]
\[ = \lim_{h \to 0} \frac{2 \sin^2 \frac{h}{2}}{\frac{\frac{4 h^2}{4}}{\frac{\sin h}{h}}}\]
\[ = \frac{2}{4}\]
\[ = \frac{1}{2}\]

 

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APPEARS IN

RD Sharma Class 11 Mathematics Textbook
Chapter 29 Limits
Q 33 | Page 80
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