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 $I:$ $\underset{x\to a}{\mathrm{l}\mathrm{i}\mathrm{m}}f\left(x\right)$ exists $=k$, but $\underset{x\to k}{\lim }g\left(x\right)$ does not exist. If $\underset{x\to a}{\mathrm{l}\mathrm{i}\mathrm{m}}g\left(f\left(x\right)\right)$ exists, then $x=a$ is a point of extrema for $y=f\left(x\right)$. If $f\left(x\right)$ is non-linear.

Statement $II:$ $\underset{x\to k}{\lim }g\left(x\right)$ does not exist, but $\underset{x\to a}{\lim }g\left(f\left(x\right)\right)$ exists, $f\left(x\right)$ will approach $k$ , when $x\to a$ through only one side.

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