Amit M Agarwal Solutions for Chapter: Limits, Exercise 7: Proficiency in 'Limits' Exercise 1

If $\underset{x\to \infty }{\lim }4x\left(\frac{\mathrm{\pi }}{4}-{\tan }^{-1}\frac{x+1}{x+2}\right)=$$y^2+4y+5,$ then $y$ can be equal to:

$\underset{x\to 0}{\lim }\frac{1-\cos \left(x^2\right)}{x^3(4^x-1)}$ is equal to:

If $f\left(x\right)=e^{\left[\cot x\right]}$, where $\left[y\right]$ represents the greatest integer less than or equal to $y,$ then:

Find the value of $\underset{x\to 0}{\mathrm{l}\mathrm{i}\mathrm{m}}\left[m\frac{\mathrm{s}\mathrm{i}\mathrm{n}x}{x}\right]$ (where $m\in I$ and $\left[.\right]$ denotes the greatest integer function).

If $\underset{x\to 0}{\mathrm{l}\mathrm{i}\mathrm{m}}{\left(1+ax+b{x}^2\right)}^{\frac{2}{x}}=e^3$, then :

For the following questions, choose the correct answer from the codes $\left(a\right), \left(b\right), \left(c\right)$ and $\left(d\right)$ defined as following:
Statement $I:$ $\underset{x\to 0}{\mathrm{l}\mathrm{i}\mathrm{m}}\frac{\sin \left(\mathrm{\pi s}\mathrm{i}{\mathrm{n}}^2\frac{x}{2}\right)}{x^2}=\mathrm{\pi }$
Statement $II:$ $\underset{x\to 0}{\mathrm{l}\mathrm{i}\mathrm{m}}\frac{\mathrm{s}\mathrm{i}\mathrm{n}x}{x}=1$

For the following questions, choose the correct answer from the codes $\left(a\right), \left(b\right), \left(c\right)$ and $\left(d\right)$ defined as following:
Statement $I:$ $\underset{x\to 0}{\mathrm{l}\mathrm{i}\mathrm{m}}\mathrm{s}\mathrm{e}{\mathrm{c}}^{-1}\left(\frac{\mathrm{s}\mathrm{i}\mathrm{n}x}{x}\right)=0$
Statement $II:$ $\underset{x\to 0}{\mathrm{l}\mathrm{i}\mathrm{m}}\left(\frac{\mathrm{s}\mathrm{i}\mathrm{n}x}{x}\right)=1$

For the following questions, choose the correct answer from the codes $\left(a\right), \left(b\right), \left(c\right)$ and $\left(d\right)$ defined as follows:
Let $a_n=2.\underset{n \mathrm{times}}{\underbrace{99...9 }}, n\in N$

Statement $I:$ $\left[\underset{n\to \infty }{\mathrm{l}\mathrm{i}\mathrm{m}}{a}_n\right]=\underset{n\to \infty }{\mathrm{l}\mathrm{i}\mathrm{m}}\left[a_n\right],$$\left[.\right]$ denotes the greatest integer function.

Statement $II:$ $\underset{n\to \infty }{\mathrm{l}\mathrm{i}\mathrm{m}}{a}_n=3$