Embibe Experts Solutions for Chapter: Limits, Exercise 1: Assam CEE 2018

If $[·]$ denotes the greatest integer function then $\underset{n\to \infty }{\lim }\frac{\left[x\right]+\left[2x\right]+\dots +\left[nx\right]}{n^2}$ is

For each $t\in \mathrm{ℝ}$ let $\left[t\right]$ denote the greatest integer less than or equal to $t$. Then $\underset{x\to 3}{\lim }\left(\right[x]-[2 x-1\left]\right)$

If $f$ is a continuous function on the real line, and $x^2+\left(f\right(x)-2)x-\sqrt{3}f\left(x\right)+2\sqrt{3}-3=0$. Then the value of $f\left(\sqrt{3}\right)$

Let for all $x>0, f\left(x\right)=\underset{n\to \infty }{\lim }n\left(x^{1/n}-1\right),$ then

Value of $\underset{y\to \infty }{\lim }(\sqrt{y+\sqrt{y+\sqrt{y}}}-\sqrt{y})$ is:

If $\underset{x\to 0}{\lim }\frac{x^a{\sin }^{b}x}{\sin x^c},a,b,c,\in R~\left\{0\right\}$ exists and has non-zero value, then

If the function $f\left(x\right)$ satisfies $\underset{x\to 2}{\lim }\frac{f\left(x\right)-1}{x^2-4}=e$ then $\underset{x\to 2}{\lim }f\left(x\right)$ is equal to

$l_n$ and $k_n$ are positive integers and $l_n+\sqrt{3}{k}_n={\left(3+\sqrt{3}\right)}^n$, then $\underset{n\to \infty }{\lim }{\tan }^{-1}\left(\frac{l_n}{k_n}\right)$ is