Question

The points of discontinuity of the function $f\left(x\right)={lim}_{n\to \infty }\frac{(2\sin x{)}^{2n}}{3^n-(2\cos x{)}^{2n}}$ are given by

• A. $r\pi \pm \frac{\pi }{12},rϵI$
• B. $r\pi \pm \frac{\pi }{6},rϵI$
• C. $r\pi \pm \frac{\pi }{3},rϵI$
• D. $r\pi \pm \pi ,rϵI$

Answer 1

The correct option is B

$r\pi \pm \frac{\pi }{6},rϵI$

The function $f\left(x\right)$ is discontinuous if $3^n-(2\cos x{)}^{2n}=0$

$\Rightarrow ({\cos }^{2}x{)}^n={\left(\frac{3}{4}\right)}^n$

$\Rightarrow co{s}^{2}x=\frac{3}{4}={\cos }^2\frac{\pi }{6}$

$\Rightarrow x=r\pi \pm \frac{\pi }{6},rϵI$