lim _{t} {rightarrow 0}left(1^{frac{1}{sin ^2 | vedclass

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

Question

(B) Let $L = \lim _{t}$ ${\rightarrow 0}\left(1^{\operatorname{cosec}^2 t}+2^{\operatorname{cosec}^2 t}+\ldots +n^{\operatorname{cosec}^2 t}\right)^{\

Vedclass · Accepted Answer

$n$

Vedclass · Answer

(B) Let $L = \lim _{t}$ ${\rightarrow 0}\left(1^{\operatorname{cosec}^2 t}+2^{\operatorname{cosec}^2 t}+\ldots +n^{\operatorname{cosec}^2 t}\right)^{\sin ^2 t}$.We can factor out the largest term $n^{\operatorname{cosec}^2 t}$ from the bracket:$L = \lim _{t}$ ${\rightarrow 0} \left[ n^{\operatorname{cosec}^2 t} \left( \left(\frac{1}{n}\right)^{\operatorname{cosec}^2 t} + \left(\frac{2}{n}\right)^{\operatorname{cosec}^2 t} + \ldots + 1 \right) \right]^{\sin ^2 t}$.$L = \lim _{t}$ ${\rightarrow 0} \left( n^{\operatorname{cosec}^2 t} \right)^{\sin ^2 t} \cdot \left( \left(\frac{1}{n}\right)^{\operatorname{cosec}^2 t} + \left(\frac{2}{n}\right)^{\operatorname{cosec}^2 t} + \ldots + 1 \right)^{\sin ^2 t}$.Since $\operatorname{cosec}^2 t \cdot \sin ^2 t = 1$,the first part becomes $n^1 = n$.For the second part,as $t \rightarrow 0$,$\operatorname{cosec}^2 t \rightarrow \infty$. Thus,for $k So,the expression becomes $n \cdot (0 + 0 + \ldots + 1)^0 = n \cdot 1^0 = n \cdot 1 = n$.

$\lim _{t}$ ${\rightarrow 0}\left(1^{\frac{1}{\sin ^2 t}}+2^{\frac{1}{\sin ^2 t}}+\ldots +n^{\frac{1}{\sin ^2 t}}\right)^{\sin ^2 t}$ is equal to $.......$

Similar Questions

The value of $\lim_{x \to 0} (\cos ax)^{\csc^2 bx}$ is

The value of the limit $\lim_{x \rightarrow \frac{\pi}{2}} \frac{4 \sqrt{2}(\sin 3x + \sin x)}{\left(2 \sin 2x \sin \frac{3x}{2} + \cos \frac{5x}{2}\right) - \left(\sqrt{2} + \sqrt{2} \cos 2x + \cos \frac{3x}{2}\right)}$ is

$\mathop {\lim }\limits_{x \to 0} \frac{{\sin {x^\circ}}}{x} = $

$\lim _{x \rightarrow 0} \left( \frac{\sin (\pi \cos ^2 x)}{x^2} \right) = $

$\lim _{x \rightarrow 0} \frac{\sin \left(\pi \cos ^2 x\right)}{x^2}$ is equal to

Vedclass Test Series

Exam Paper Generator

Online Exam Module