A Log Tangent Product

Let $f(x) = \ln x - \dfrac{1}{x}$ and $g(x) = ax + b$.

1. If $h(x) = f(x) - g(x)$ is increasing on $(0, +\infty)$, find the range of possible values of $a$. 2. If the line $y = ax + b$ is tangent to the graph of $f(x)$, find the minimum value of $a + b$. 3. When $b = 0$, suppose the graphs of $f(x)$ and $g(x)$ meet at two points $A(x_1, y_1)$ and $B(x_2, y_2)$. Prove that $x_1 x_2 > 2e^2$.