$\mathrm{I} : \underset{\theta \to 0}{lim} \left(\left[\frac{n\cdot \sin \theta }{\theta }\right]+\left[\frac{n\tan \theta }{\theta }\right]\right)=$ even integer where [.] denotes the greatest integer function

$\text{ II : }\underset{x\to 0}{lim} \frac{a{e}^x-b}{x}=2\text{ then }a=2,b=2$

$\left(I\right)$

$\underset{\theta \to 0}{lim} \left[\frac{n\sin \theta }{\theta }\right]=\mathrm{n}-1\dots ..[\text{ from graph maximum value of }\sin \mathrm{x}/\mathrm{x}=1]$

$\underset{\theta \to 0}{lim} \left[\frac{n\tan \theta }{\theta }\right]=n\dots ...[\text{ from graph minimum value of }\tan x/x=1]$

$\underset{x\to \theta }{lim} \left(\left[\frac{n\sin \theta }{\theta }\right]+\left[\frac{n\tan \theta }{\theta }\right]\right)=\mathrm{n}-1+\mathrm{n}=2\mathrm{n}-1$

$\left(II\right)$

$$\begin{gathered} \underset{x\to 0}{lim} \frac{a{e}^x-b}{x}=2 \\ a-b=2 \\ \underset{x\to 0}{lim} \frac{a{e}^x-b}{x}=2 \\ Apply L\text{'}hospital rule \\ a=2 \\ a=2,b=0 \end{gathered}$$