Unbounded Despite Appearances

Let $f(x) = \dfrac{\mathrm{e}^x}{x^2 + 1}$ be defined on $(0, +\infty)$.

1. Prove that $f(x) > 1$. 2. Let $g(x) = \dfrac{\mathrm{e}^x - 1}{x}$. Find the intervals on which $g(x)$ is monotonic, and prove that for every positive real number $m$ there exists $x_0 > 0$ with $g(x_0) > m$.