Embibe Experts Solutions for Chapter: Limits, Exercise 2: EXERCISE-2

Which one of the following best represents the graph of the function $f\left(x\right)=\underset{n\to \infty }{\lim }\frac{2}{\pi }{\tan }^{-1}\left(nx\right)$

$\underset{x\to 0}{\lim }\frac{\left(\sqrt{\left(1-\cos x\right)+\sqrt{\left(1-\cos x\right)+\sqrt{\left(1-\cos x\right)+\mathrm{.........}\infty }}}\right)-1}{x^2}$

Let $a,b,c$ are non zero constant number then $\underset{r\to \infty }{\lim }\left[\frac{\cos \left({\displaystyle \frac{a}{r}}\right)-\cos \left({\displaystyle \frac{b}{r}}\right)\cos \left({\displaystyle \frac{c}{r}}\right)}{\sin \left({\displaystyle \frac{b}{r}}\right)\sin \left({\displaystyle \frac{c}{r}}\right)}\right]$ equals.

$\underset{n\to \infty }{\lim }\left[{\left(\frac{n}{n}\right)}^n+{\left(\frac{n-1}{n}\right)}^n+\mathrm{.....}+{\left(\frac{1}{n}\right)}^n\right]$ equals-

Given $l_1=\underset{x\to \frac{\pi }{4}}{\lim }{\cos }^{-1}\left[\sec \left(x-\frac{\pi }{4}\right)\right]; l_2=\underset{x\to \frac{\pi }{4}}{\lim }{\sin }^{-1}\left[\mathrm{cosec}\left(x+\frac{\pi }{4}\right)\right]$;

$l_3=\underset{x\to \frac{\pi }{4}}{\lim }{\tan }^{-1}\left[\cot \left(x+\frac{\pi }{4}\right)\right]; l_4=\underset{x\to \frac{\pi }{4}}{\lim }{\cot }^{-1}\left[\tan \left(x-\frac{\pi }{4}\right)\right]$

where $\left[x\right]$ denotes greatest integer function then which of the following limits exist?

$\underset{n\to \infty }{\lim }\cos \left(\pi \sqrt{n^2+n}\right)$ when $n$ is an integer-

$\underset{x\to 0^{+}}{\lim }\left[\frac{1}{x\sqrt{x}}\left(a\arctan \left(\frac{\sqrt{x}}{a}\right)-b\arctan \left(\frac{\sqrt{x}}{b}\right)\right)\right]$ has the value equal to-

The value of $\underset{x\to 0}{\lim }\left(\frac{\sqrt[3]{1+\sin x}-\sqrt[3]{1-\sin x}}{x}\right)$ is