Final answer:
The question involves evaluating the convergence or divergence of certain integrals related to probability functions, but lacks sufficient context to provide direct answers. Convergent integrals approach a finite limit, while divergent ones do not. Probability functions must converge to a total probability of 1 over their range.
Step-by-step explanation:
The provided question appears to involve the evaluation of definite integrals and understanding of continuous probability density functions in the context of mathematics, particularly calculus. The student is asking for assistance in determining whether certain improper integrals and limits are convergent or divergent. Unfortunately, the question as presented contains some disjointed passages and references that don't relate directly to the functions F(x) and G(x) as defined. We need additional context or a clearer statement of the problem to directly address the convergence or divergence of specific integrals mentioned.
However, to clarify, a function or integral is said to be convergent if it approaches a finite limit as the variable approaches infinity or as the interval of integration is extended indefinitely. On the other hand, a function or integral is divergent if it does not approach a finite limit under said conditions.
Considering continuous probability distribution functions, as vaguely hinted at in the question, we can state that a continuous probability density function is a function f(x), such that the integral of f(x) from a to b gives the probability of the random variable falling between a and b. In all cases, if f(x) is a probability density function, the integral from negative to positive infinity should equal 1. This is important when discussing the convergence of probability functions, as any probability function must be normalized to converge to 1 over its entire range.