Embibe Experts Solutions for Exercise 8: Assignment

If $\underset{x\to y}{\lim }\frac{{\sin }^{2}x-{\sin }^{2}y}{x^2-y^2}=g\left(y\right)$ and the limiting value of $\underset{y\to 0}{\lim }g\left(y\right)=k$, then the value of $1+k+k^2+k^3$ $+\dots \dots +k^{2010}$ is

Let $f\left(x\right)=\frac{x+x^2+\dots \dots +x^n-n}{x-1},g\left(x\right)={\left(4^n+5^n\right)}^{\frac{1}{n}}$. and $\alpha$ and $\beta$ are the roots of equation $\underset{x\to 1}{\lim }f\left(x\right)$ $=\underset{n\to \infty }{\lim }g\left(x\right)$, then the value of$\sum _{m=0}^{\infty }{\left(\frac{1}{\alpha }+\frac{1}{\beta }\right)}^m$ is

The value of $\underset{x\to 0}{\lim }\left\{\left[\frac{2018\sin x}{x}\right]+\left[\frac{2019\tan x}{x}\right]\right\}$, (where $\left[x\right]$ denotes the greatest integer $\le x$) is

If ${Lt}_{x\to 1}{\left(1+ax+b{x}^2\right)}^{\frac{c}{x-1}}=e^3$, then the value of $a+b+bc+2013$ is

Let $f\left(x\right)=\cos \left(\frac{x}{2}\right)·\cos \left(\frac{x}{4}\right).....\cos \left(\frac{x}{2^n}\right)$. If $\underset{n\to \infty }{\lim }f\left(x\right)=g\left(x\right)$, then the value of$\underset{x\to 0}{\lim }\left(\frac{1-{\left(g\left(x\right)\right)}^{2011}}{1-g\left(x\right)}\right)$ is

$\underset{x\to 0}{\lim }\left(1+ax\right){}^{\frac{b}{x}}=e^2$, where $a,b\in N$, such that $a+b=3$, then the value of $\left(a, b\right)$ is

$\underset{x\to 1}{\lim }\left[{\sec }^{-1}\left(\frac{{\lambda }^2}{\mathrm{In}x}-\frac{{\lambda }^2}{x-1}\right)\right]$ exists, then $\lambda$ belongs to 

Let $f\left(x\right)=\frac{\left(x-1\right)}{\left(2x^2-7x+5\right)}$ then