Question

Evaluate: $\underset{x\to \pi /3}{lim}\frac{\cos (x+\frac{\pi }{6})}{{(1-2\cos x)}^{2/3}}$

• A. $1$
• B. $-1$
• C. $0$
• D. None of these

Answer 1

The correct option is C $0$

$\underset{x\to \frac{\pi }{3}}{lim}\frac{\cos (x+\frac{\pi }{6})}{{(1-2\cos x)}^{\frac{2}{3}}}=\underset{z\to 0}{lim}\frac{\cos (\frac{\pi }{2}+z)}{{[1-2\cos (\frac{\pi }{3}+z)]}^{\frac{2}{3}}}$ $[$ Substituting $x-\frac{\pi }{3}=z]$

$=\underset{z\to 0}{lim}\frac{-\sin z}{{(1-\cos z+\sqrt{3}\sin z)}^{\frac{2}{3}}}$

$=\underset{z\to 0}{lim}\frac{-2\sin \left(\frac{z}{2}\right)\cos \left(\frac{z}{2}\right)}{{\left[2\sin \left(\frac{z}{2}\right)\right]}^{\frac{2}{3}}{\left[\sin \left(\frac{z}{2}\right)+\sqrt{3}\cos \left(\frac{z}{2}\right)\right]}^{\frac{2}{3}}}$

$=\underset{z\to 0}{lim}\frac{{-2}^{\frac{1}{3}}{\left[\sin \left(\frac{z}{3}\right)\right]}^{\frac{1}{3}}\cos \left(\frac{z}{2}\right)}{{[\sin \left(\frac{z}{2}\right)+\sqrt{3}\cos \left(\frac{z}{2}\right)]}^{\frac{2}{3}}}=\frac{{-2}^{\frac{1}{3}}}{{\left(\sqrt{3}\right)}^{\frac{2}{3}}}=0$