Prove inequality for tail of normal distribution

Question

I have to prove this inequality for $x>0$. I have no idea how to even start this. I would appreciate any help. $$\frac{x^{2}}{1+x^{2}}\frac{1}{x}\exp \left( \frac{-x^{2}}{2}\right)\leqslant \int_{x}^{\infty}e^{-t^{2}/2}dt\leqslant \frac{1}{x}\exp\left(\frac{-x^{2}}{2}\right)$$ I tried to calculate derivatives of two sites but I don't think this is the right way to solve this problem.

Answer 1

For the upper bound use the inequality

$$\int_x^{\infty} e^{-t^2/2} \, dt \leq \int_x^{\infty} \frac{t}{x} e^{-t^2/2} \, dt.$$

For the lower bound note that

$$\frac{1}{x^2} \int_x^{\infty} e^{-t^2/2} \, dt \geq \int_x^{\infty} \frac{1}{t^2} e^{-t^2} \, dt$$

and apply the integration by parts formula.