# Lim X → π / 3 Sin ( π 3 − X ) 2 Cos X − 1 is Equal to - Mathematics

MCQ

$\lim_{x \to \pi/3} \frac{\sin \left( \frac{\pi}{3} - x \right)}{2 \cos x - 1}$ is equal to

#### Options

• $\sqrt{3}$

• $\frac{1}{2}$

• $\frac{1}{\sqrt{3}}$

• $\sqrt{3}$

#### Solution

$\frac{1}{\sqrt{3}}$

$\lim_{x \to \frac{\pi}{3}} \frac{\sin \left( \frac{\pi}{3} - x \right)}{2 \cos x - 1}$
$= \lim_{h \to 0} \frac{\sin \frac{\pi}{3} - \left( \frac{\pi}{3} - h \right)\left[ \right]}{2 \cos \left( \frac{\pi}{3} - h \right) - 1}$
$= \lim_{h \to 0} \frac{\sin h}{2\left[ \cos \frac{\pi}{3}\cos h + \sin \frac{\pi}{3} \sin h \right] - 1}$
$= \lim_{h \to 0} \frac{\sin h}{2\left[ \frac{1}{2}\cos h + \frac{\sqrt{3}}{2} \sin h \right] - 1}$
$= \lim_{h \to 0} \frac{\sin h}{\cos h + \sqrt{3} \sin h - 1}$
$= \lim_{h \to 0} \frac{\sin h}{- 2 \sin^2 \frac{h}{2} + \sqrt{3} \sin h}$
$\text{ Dividing } N^r \text{ and } D^r \text{ by } h:$
$= \lim_{h \to 0} \frac{\frac{\sin h}{h}}{- \left( 2 \times \frac{h}{4} \right) \left( \frac{\sin^2 \frac{h}{2}}{h \times \frac{h}{4}} \right) + \frac{\sqrt{3} \sin h}{h}}$
$= \frac{1}{\sqrt{3}}$

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#### APPEARS IN

RD Sharma Class 11 Mathematics Textbook
Chapter 29 Limits
Q 21 | Page 79