An Area Ratio Minimum

The hyperbola $C \colon \dfrac{x^2}{4} - y^2 = 1$ has left and right vertices $A_1, A_2$. A line $l$ through $P(4, 0)$ meets the right branch of $C$ at two points $M$ and $N$.

1. If the slope $k$ of $l$ exists, find the range of possible values of $k$. 2. Let $k_1, k_2$ be the slopes of $A_1M$ and $A_2N$. Find the value of $\dfrac{k_1}{k_2}$. 3. Let $G$ be the intersection of lines $A_1M$ and $A_2N$, and let $S_1, S_2$ be the areas of triangles $GMN$ and $GA_1A_2$. Find the minimum value of $\dfrac{S_1}{S_2}$.