Bounds on function $\exp(-\frac{1}{2}x^2)$

Question

I have the following function : $$f(x)=\exp(-\frac{1}{2}x^2),$$ where $x >0$.

I am looking for some tight bounds (upper bound and lower bounds) on $f(x)$. Any idea ?

P.S.: The problem arises when I tried to find an upper bound on $x_2 \exp(-x_1^2/2)/ (x_1 \exp(-x_2^2/2))$. I prefer the bounds to be tight so that the error when bounding will be small.

Answer 1

We have $$ |f(x)|=\left|\exp(-\frac{1}{2}x^2)\right|<1, \quad x>0, $$ and $$ \lim_{x \to +\infty}|f(x)|=\lim_{x \to +\infty}\left|\exp(-\frac{1}{2}x^2)\right|=0. $$

Answer 2

When $x=0$ $f(x)=1$ and it decreases from there to zero in the limit. So $0$ and $1$ are bounds.