$$\begin{aligned} &\text { Given, } \lim _{x \rightarrow 0}\left(\sin m x \cot \frac{x}{\sqrt{3}}\right)=2\\ &\begin{aligned} & \Rightarrow \quad \lim _{x \rightarrow 0} \frac{\sin m x}{m x} \cdot m x \cdot \frac{1}{\tan \frac{x}{\sqrt{3}}}=2 \\ & \Rightarrow \quad \lim _{x \rightarrow 0} \frac{\sin m x}{m x} \cdot m x \cdot \frac{\frac{x}{\sqrt{3}}}{\tan \frac{x}{\sqrt{3}}} \cdot \frac{\frac{1}{x}}{\frac{x}{\sqrt{3}}}=2 \\ & \Rightarrow \quad \lim _{x \rightarrow 0} \frac{\sin m x}{m x} \cdot \lim _{x \rightarrow 0} \frac{\frac{x}{\sqrt{3}}}{\tan \frac{x}{\sqrt{3}}} \cdot \lim _{x \rightarrow 0} \frac{m x}{\frac{x}{\sqrt{3}}}=2 \\ & \Rightarrow \quad \sqrt{3} x=2 \\ \therefore\quad & m=\frac{2 \sqrt{3}}{3} \end{aligned} \end{aligned}$$

$$\begin{aligned} \text{Given,}\quad y & =1+\frac{x}{1!}+\frac{x^2}{2!}+\frac{x^3}{3!}+\frac{x^4}{4!}+\ldots \\ \therefore\quad \frac{d y}{d x} & =0+1+\frac{2 x}{2}+\frac{3 x^2}{6}+\frac{4 x^3}{4!} \\ & =1+x+\frac{x^2}{2}+\frac{x^3}{6}+\ldots \\ & =1+\frac{x}{1!}+\frac{x^2}{2!}+\frac{x^3}{3!}+\ldots \\ & =y \end{aligned}$$

Given,

$$\begin{aligned} \lim _{x \rightarrow 3^{+}} \frac{x}{[x]} & =\lim _{h \rightarrow 0} \frac{(3+h)}{[3+h]} \\ & =\lim _{h \rightarrow 0} \frac{(3+h)}{3}=1 \end{aligned}$$