JEE Math Practice Question

If $$\lim\limits_{x \rightarrow 0} \frac{\alpha \mathrm{e}^{x}+\beta \mathrm{e}^{-x}+\gamma \sin x}{x \sin ^{2} x}=\frac{2}{3}$$, where $$\alpha, \beta, \gamma \in \mathbf{R}$$, then which of the following is NOT correct?

1. A.$$\alpha^{2}+\beta^{2}+\gamma^{2}=6$$
2. B.$$\alpha \beta+\beta \gamma+\gamma \alpha+1=0$$
3. C.$$\alpha\beta^{2}+\beta \gamma^{2}+\gamma \alpha^{2}+3=0$$
4. D.$$\alpha^{2}-\beta^{2}+\gamma^{2}=4$$