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Q.
The set of points of discontinuity of the function $f ( x )$, where $f(x)=\displaystyle\lim _{n \rightarrow \infty} \frac{(2 \sin x)^{2 n}}{3^{n}-(2 \cos x)^{2 n}}$ is
The function is discontinuous when the denominator is $0$
$3^{n}-(2 \cos x)^{2 n}=0$
$3^{n}=(2 \cos x)^{2 n}$
$3^{ n }=\left(4 \cos ^{2} x \right)^{ n }$
$3=4 \cos ^{2} x$
$\cos ^{2} x=\frac{3}{4}$
Thus, at every $n \pi \pm \frac{\pi}{6}$ the function is discontinuous where $n$ belongs to integer.