The set of points of discontinuity of the function f ( x ), where f(x)= displaystyle lim n arrow ∞ ((2 sin x)2 n/3n-(2 cos x)2 n) is

Question

The set of points of discontinuity of the function f ( x ), where f(x)= displaystyle lim n arrow ∞ ((2 sin x)2 n/3n-(2 cos x)2 n) is (A) R (B)  n π ∓ (π/3), n ε I  (C)  n π ± (π/6), n ε I  (D) None of these

Tardigrade Admin · Accepted Answer

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} = (4 cos^{2} x)^{n}

3 = 4 cos^{2} x

cos^{2} x = \frac{3}{4}

Thus, at every

nπ \pm \frac{π}{6}

the function is discontinuous where

n

belongs to integer.

Q. The set of points of discontinuity of the function $f (x)$ , where $f (x) = n \to \infty lim \frac{( 2 sin x ) ^{2 n}}{3 ^{n} - ( 2 cos x ) ^{2 n}}$ is

$R$

${nπ \mp \frac{π}{3}, n ε I}$

${nπ \pm \frac{π}{6}, n ε I}$

None of these

The function is discontinuous when the denominator is $0$
$3^{n} - (2 cos x)^{2 n} = 0$
$3^{n} = (2 cos x)^{2 n}$
$3^{n} = (4 cos^{2} x)^{n}$
$3 = 4 cos^{2} x$
$cos^{2} x = \frac{3}{4}$
Thus, at every $nπ \pm \frac{π}{6}$ the function is discontinuous where $n$ belongs to integer.

Q. The set of points of discontinuity of the function f(x), where f(x)=n→∞lim​3n−(2cosx)2n(2sinx)2n​ is

R

{nπ∓3π​,nεI}

{nπ±6π​,nεI}

None of these

The function is discontinuous when the denominator is 0 3n−(2cosx)2n=0 3n=(2cosx)2n 3n=(4cos2x)n 3=4cos2x cos2x=43​ Thus, at every nπ±6π​ the function is discontinuous where n belongs to integer.

Q. The set of points of discontinuity of the function $f (x)$ , where $f (x) = n \to \infty lim \frac{( 2 sin x ) ^{2 n}}{3 ^{n} - ( 2 cos x ) ^{2 n}}$ is

$R$

${nπ \mp \frac{π}{3}, n ε I}$

${nπ \pm \frac{π}{6}, n ε I}$

The function is discontinuous when the denominator is $0$
$3^{n} - (2 cos x)^{2 n} = 0$
$3^{n} = (2 cos x)^{2 n}$
$3^{n} = (4 cos^{2} x)^{n}$
$3 = 4 cos^{2} x$
$cos^{2} x = \frac{3}{4}$
Thus, at every $nπ \pm \frac{π}{6}$ the function is discontinuous where $n$ belongs to integer.