The value of lim _{n rightarrow infty} n sin | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(B) Let $L = \lim _{n ightarrow \infty} n \sin \frac{2 \pi}{3 n} \cos \frac{2 \pi}{3 n}$.We know that $\sin(2	heta) = 2 \sin 	heta \cos 	heta$,so

Vedclass · Accepted Answer

$\frac{2 \pi}{3}$

Vedclass · Answer

(B) Let $L = \lim _{n ightarrow \infty} n \sin \frac{2 \pi}{3 n} \cos \frac{2 \pi}{3 n}$.We know that $\sin(2	heta) = 2 \sin 	heta \cos 	heta$,so $\sin 	heta \cos 	heta = \frac{1}{2} \sin(2	heta)$.Here,$	heta = \frac{2 \pi}{3 n}$.Thus,$L = \lim _{n ightarrow \infty} n \cdot \frac{1}{2} \sin \left( 2 \cdot \frac{2 \pi}{3 n} ight) = \lim _{n ightarrow \infty} \frac{n}{2} \sin \left( \frac{4 \pi}{3 n} ight)$.Using the standard limit $\lim _{x ightarrow 0} \frac{\sin x}{x} = 1$,we rewrite the expression:$L = \lim _{n ightarrow \infty} \frac{n}{2} \cdot \left( \frac{4 \pi}{3 n} ight) \cdot \frac{\sin \left( \frac{4 \pi}{3 n} ight)}{\left( \frac{4 \pi}{3 n} ight)}$.As $n ightarrow \infty$,$\frac{4 \pi}{3 n} ightarrow 0$,so $\frac{\sin \left( \frac{4 \pi}{3 n} ight)}{\left( \frac{4 \pi}{3 n} ight)} ightarrow 1$.Therefore,$L = \lim _{n ightarrow \infty} \frac{n}{2} \cdot \frac{4 \pi}{3 n} \cdot 1 = \frac{4 \pi}{6} = \frac{2 \pi}{3}$.

The value of $\lim _{n \rightarrow \infty} n \sin \frac{2 \pi}{3 n} \cos \frac{2 \pi}{3 n}$ is

Similar Questions

$\mathop {{\rm{lim}}}\limits_{x \to 0} \frac{{\left( {1 - \cos 2x} \right)\left( {3 + \cos x} \right)}}{{x\tan 4x}} = $

$\mathop {\lim }\limits_{x \to 3} \frac{{\sqrt {3x} - 3}}{{\sqrt {2x - 4} - \sqrt 2 }}$ is equal to

$\lim _{x \rightarrow 0} \left( \frac{\tan x}{\sqrt{2x+4}-2} \right)$ is equal to

Evaluate the limit: $\mathop {\lim }\limits_{x \to \infty } [x({a^{1/x}} - 1)]$,where $a > 1$.

$\mathop {\lim }\limits_{x \to 0} \frac{{\sin ({x^{1/3}})\ln (1 + 3x)}}{{{{(\tan^{ - 1}\sqrt x )}^2}({e^{5{x^{1/3}}}} - 1)}} = $

Vedclass Test Series

Exam Paper Generator

Online Exam Module