If \( \displaystyle \lim_{x\to 0}\dfrac{\alpha e^{x}+\beta e^{-x}+\gamma \sin x}{x\sin^{2}x}=\dfrac{2}{3} \), where \( \alpha,\beta,\gamma\in \mathbb{R} \), then which of the following is NOT correct?
\( \alpha^{2}+\beta^{2}+\gamma^{2}=6 \)
\( \alpha\beta+\beta\gamma+\gamma\alpha+1=0 \)
\( \alpha\beta^{2}+\beta\gamma^{2}+\gamma\alpha^{2}+3=0 \)
\( \alpha^{2}-\beta^{2}+\gamma^{2}=4 \)
