Embibe Experts Solutions for Chapter: Limits, Exercise 1: Exercise 1

Let $n$ be an odd integer. If $\sin n\theta =\sum _{r=0}^n{b}_r{\sin }^r\theta$, for every value of $\theta$, then -

If $f\left(x\right)=|x{|}^{\sin x},$ then

If $\underset{x\to 0^{+}}{\lim }{\log }_{\sin \frac{x}{2}}\sin x$ is equal to

$\underset{n\to \infty }{\lim }{\left({\left(\frac{n}{n+1}\right)}^{\alpha }+\sin \frac{1}{n}\right)}^n$ 

Find the value of $\underset{x\to 0}{\lim }{\log }_{{\tan }^{2}x}\left({\tan }^{2}2x\right)$

Find the value of $\underset{x\to 0}{\lim }\left[\frac{1}{x{\sin }^{-1}x}-\frac{1-x^2}{x^2}\right]$

If $\underset{x\to 0}{\lim }\left[\frac{1-\cos 3x·\cos 9x·\cos 27x.....\cos 3^{n}x}{1-\cos {\displaystyle \left(\frac{x}{3}\right)}\cos {\displaystyle \left(\frac{x}{9}\right)}\cos {\displaystyle \left(\frac{x}{27}\right)}....\cos {\displaystyle \left(\frac{1}{3^n}\right)}x}\right]=3^{10}$, find the value of $n$.

If $\underset{x\to 0}{\lim }\left[\frac{1-\cos x\cos 2x\cos 3x....\cos nx}{x^2}\right]$ has the value equal to $253$, find the value of $n$ (where $n\in N$).