Prove that for \(x>0\) the function \(\int_{x}^{\infty} e^{-\frac{z^{z}}{2}} d z\) satisfies the inequalities

\[\frac{x}{1+x^{2}} e^{-\frac{1}{2} x^{2}} \leqslant \int_{x}^{\infty} e^{-\frac{1}{2} z^{2}} d z \leqslant \frac{1}{x} e^{-\frac{1}{2} x^{3}}\]