Convergance and Laplace transforms

[heatgrad]

Hi,
Is the final value theorem an appropriate way to deduce if a specific differential equations converges or diverges?

 

[heatgrad]

What I guess I mean is, does it offer the predict the same results as finding positive or negative real roots?

I've tried it out for myself a few times and my own results are...inconclusive. But I'm kind of getting back into this method of dealing with ODEs after letting it lie for over a year. And finding literature about it largely proves unfruitful, especially because describing the information I want includes very common terms or too many terms, causing a tidal wave of unwanted results.