I made an interesting observation about the Laplace transform that I think might pose a nice challenge, if hopefully there isn't anything obvious that I missed.
Recall the final value theorem (FVT) : $\forall f$ bounded on $(0,\infty)$ and such that $\lim\limits_{t\to\infty}f(t)$ is finite, $$\lim\limits_{s\to0^+}\left[s\mathcal{L}\left\lbrace f\right\rbrace(s)\right]=\lim\limits_{t\to\infty}f(t)$$ with $\displaystyle\mathcal{L}\left\lbrace f\right\rbrace(s)=\int_{0}^{\infty}e^{-st}f(t)dt$ the Laplace transform of $f$.
Now, I made the following observation :
For a significant number of functions $f$, the limit $$\lim\limits_{s\to0^+}\frac{s}{f\left(\frac{1}{s}\right)}\mathcal{L}\left\lbrace f\right\rbrace(s)$$ is finite.
We should note that, due to the FVT theorem stated above, it is obviously true when $f$ satisfies the above requirements for the FVT.
The thing is, this result, although weaker than the FVT when the FVT applies, seem to apply to a much wider range of function.
So far, I have tested it on polynomial functions, hyperbolic functions, exponential and logarithmic functions. This results always seems to holds whenever the function is continuous, and either monotonically increasing and concave, or monotonically decreasing and convex on $(0,\infty)$.
For instance, for $f:x\mapsto \sqrt{x}$ (which is neither bounded nor has a finite limit at $\infty$), the limit is $\frac{\sqrt{\pi}}{2}$ ; and for $f=\log_b$ for any base $b$ (including the natural log), the limit is $1$. But, as metamorphy pointed out in a comment, it doesn't work for $f:x\mapsto e^{\sqrt{x}}$ : that's because this function is monotonically increasing but not concave on $(0,\infty)$.
So here's my conjecture :
Show that $\forall f$ continuous, and either monotonically increasing and concave, or monotonically decreasing and convex on $(0,\infty)$, $$\lim\limits_{s\to0^+}\frac{s}{f\left(\frac{1}{s}\right)}\mathcal{L}\left\lbrace f\right\rbrace(s)$$ is finite.
Wether or not the requirements for this result to holds are correct or not, I also wish to find the value of this limit.
I don't have a clue as how to proceed to tackle this. Any insight would be much appreciated.