A Second Crossing

Let $f(x) = \mathrm{e}^x - \dfrac{x^2}{2}$, and let $l$ be the tangent line to $y = f(x)$ at the point $P(x_0, f(x_0))$ with $x_0 > 0$.

1. Prove that $l$ meets the curve $y = f(x)$ at exactly one point $(x_1, f(x_1))$ other than $P$, and that $x_1 < 0$. 2. Under the conditions of part 1, find the range of possible values of $\dfrac{x_0}{x_1}$.