Chọn A

$A=\left\{ \left. x\in \mathbb{R} \right|x+3<4+2x \right\}$$\Rightarrow A=\left( -1;\,+\infty  \right).$

$B=\text{ }\!\!\{\!\!\text{ }\left. x\in \mathbb{R} \right|5x-3<7\text{ }\!\!\}\!\!\text{ }$

$A\cap B=\left( -1;\,2 \right)$ $\Leftrightarrow A\cap B=\left\{ \left. x\in \mathbb{R}\, \right|-1\,< \,x\, < \,\,2 \right\}.$.

$\Rightarrow A\cap B=\left\{ \left. x\in \mathbb{N} \right|\,-1\,\,< \,\,x\,\, <\,\,2 \right\}$ $\Leftrightarrow A\cap B=\left\{ 0;1 \right\}.$