How to determine the behavior of a function by analyzing it's Laplace transform?

Question

In an exercise, I'm given the Laplace transform of a function $f(t):$ $$\mathcal{L}\{y\}(s)=\frac{9}{s(s^2 + 3s + 1)}$$

and then the teacher asks the following question:

What is the behavior of the function $y$?

1. The function oscillates but doesn't converge
2. The function oscillates and converges to 3
3. The function oscillates and converges to 9
4. The function converges to 3 without any oscillations
5. The function converges to 9 without any oscillations

I tried to find an explicit formula for $y$ by trying to reverse the laplace transform but I wasn't able to do so. That got me thinking that maybe there is a way to solve this just by analyzing the laplace transform itself. I tried to look on my class notes for something that might help but I couldn't find anything, and on the internet I only found the following result:

$$\lim_{s\to\infty} s \ \mathcal{L}\{y\}(s) = y(0^+)$$

Is there a way to determine the behavior of $y$ just by analyzing its laplace transform? If so, are there any more important results like the one above (that relate the laplcae transform to the original function) that I should be aware of?

Answer 1

There is a property known as the "Final Value Theorem," that states

$$\lim_{t\to \infty}f(t)=\lim_{s\to 0}sF(s)$$

And the roots of the denominator are all real.

Can you finish now?