asymptotics - Behaviour and Limit of Cumulative Distribution as Variance Grows to Infinity

Please suggest:

What are some reasonable assumptions regarding the limit of the Cumulative Distribution as the Variance grows to infinity.

$$ \lim_{\sigma\rightarrow\infty} F\left(t,\sigma\right) = \text{??} $$

Also, is it a reasonable assumption to expect that the cumulative distribution will decrease in value with growing variance, as shown below?

$$\frac{\partial F(t,\sigma)}{\partial\sigma}\leq0$$

Here, $F(t,\sigma)$ are a family of distributions with parameter governed by $\sigma$, the variance. If helpful, we can make another simplifying assumption that $t\geq0$.

Please note, the questions are not specific to any particular distribution, but any general distribution and what are some valid forms of this limit and the behaviour of the distribution as variance increases.

Please point out any specific references on this topic and also list any additional assumptions made to arrive at any results.

Please let me know if the question is not clear or if you need any further information.