Question

The set of points of discontinuity of the function $f ( x )=\displaystyle\lim _{ n \rightarrow \infty} \frac{(2 \sin x )^{2 n }}{3^{ n }-(2 \cos x )^{2 n }}$ is given by

• A. $R$
• B. $\left\{ n \pi \pm \frac{\pi}{3}, n \in I \right\}$
• C. $\left\{ n \pi \pm \frac{\pi}{6}, n \in I \right\}$
• D. None of these

Answer 1

Answer: (C)

We have, $f(x)=\displaystyle\lim _{n \rightarrow \infty} \frac{(2 \sin x)^{2 n}}{3^{n}-(2 \cos x)^{2 n}}$
$=\displaystyle\lim _{n \rightarrow \infty} \frac{(2 \sin x)^{2 n}}{(\sqrt{3})^{2 n}-(2 \cos x)^{2 n}}$
$f ( x )$ is discontinuous when $(\sqrt{3})^{2 n }-(2 \cos x )^{2 n }=0$
i.e. $\cos x=\pm \frac{\sqrt{3}}{2} \Rightarrow x=n \pi \pm \frac{\pi}{6}(n \in I )$