displaystyle lim n arrow ∞ n sin (2 π/3 n) ⋅ cos (2 π/3 n) is

Question

displaystyle lim n arrow ∞ n sin (2 π/3 n) ⋅ cos (2 π/3 n) is (A) 1 (B)  (π /3) (C)  (π/6) (D)  (2π/3)

Tardigrade Admin · Accepted Answer

displaystyle lim n arrow ∞ n ⋅ sin (2 π/3 n) ⋅ cos (2 π/3 n) = displaystyle lim n arrow ∞ n (( sin (2 π/3 n))/((2 π)3 n)) ⋅ cos (2 π/3 n) × (2 π/3 n) =(1) ⋅ cos (0°) × (2 π/3) ∵ displaystyle lim θ arrow ∞ ( sin 1 / θ/1 / θ)=1 =1 ⋅ (2 π/3)=(2 π/3)

Q. $n \to \infty lim n sin \frac{2 π}{3 n} \cdot cos \frac{2 π}{3 n}$ is

$1$

$\frac{π}{3}$

$\frac{π}{6}$

$\frac{2 π}{3}$

Q. n→∞lim​nsin3n2π​⋅cos3n2π​ is

1

3π​

6π​

32π​

n→∞lim​n⋅sin3n2π​⋅cos3n2π​ =n→∞lim​n{(3n2π​)(sin3n2π​)​}⋅cos3n2π​×3n2π​ =(1)⋅cos(0∘)×32π​ {∵θ→∞lim​1/θsin1/θ​=1} =1⋅32π​=32π​

Q. $n \to \infty lim n sin \frac{2 π}{3 n} \cdot cos \frac{2 π}{3 n}$ is

$1$

$\frac{π}{3}$

$\frac{π}{6}$

$\frac{2 π}{3}$