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

$\underset{x\to 0}{\lim }\left(\frac{{\sin }^{-1}x-{\tan }^{-1}x}{x^3}+\frac{84x{\tan }^{-1}\left(\sqrt{2}-1\right)}{\sin \pi x}\right)$ is equal to

If $\underset{x\to 0}{\lim }\frac{x^3}{\sqrt{a+x}(bx-\sin x)}=1,$ then the value of $(a+b)$ where $a>0,$ is

If $f\left(x\right)=\sum _{\lambda =1}^n\left(x-\frac{5}{\lambda }\right)\left(x-\frac{4}{\lambda +1}\right)$, then $\underset{n\to \infty }{\lim }f\left(0\right)$ is equal to

Let $f\left(x\right)=\left\{\begin{array}{rll}(-1{)}^{\left[x^2\right]}& \text{if}& x<0\\ \underset{n\to \infty }{\lim }\frac{1}{1+x^n}& \text{if}& x\ge 0\end{array}\right..$ Then $\underset{x\to 0^{-}}{\lim }5f\left(x\right)+\underset{\mathrm{x}\to 0^{+}}{\lim }7f\left(x\right)$ equals (where [.] represents the greatest integer function)

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

If $\underset{x\to 0}{\lim }{\left(\cos x+a\sin bx\right)}^{\frac{1}{\mathrm{x}}}=e^2,$ then the possible values of $a \& b$ are

If $\underset{x\to 0}{\lim }{\left(1+ax+b{x}^2\right)}^{\frac{2}{x}}=e^3,$ then possible values of $a$ and $b$ is/are

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