Let $g_k(x)$ be a decreasing function respect to $x$ defined in $\mathbb{R}$ which satisfies the condition $$ \frac{g_k(y)}{g_k(z)} \to \infty \text{ when } k \to \infty \quad \text{and whenever } \, y<z. $$ Can we deduce anything about the limit $g_k(x) \to \infty$ as $k \to \infty$?
Maybe that this limit is almost always $0$ or $\infty$?
Or maybe if we add some additional conditions we can say something? I’m a bit unsure about this.
Let $x_0$ be such that $\lim_{k\to\infty}g_k(x_0) = A$ and $A$ is not $0$ nor $\infty$. Then by your condition for any other $x$ we have either $g_k(x)/g_k(x_0) \to\infty$ or $g_k(x)/g_k(x_0)\to 0$ as $k\to\infty$. But in both cases we have $$ \lim_{k\to\infty} \frac{g_k(x)}{g_k(x_0)} = \frac{1}{A} \lim_{k\to\infty}{g_k(x)}. $$ We can make a conclusion now.