Are Laplace Transform Limits Equivalent to a Limit at Infinity?

[matematikuvol]

How we get relation
$$\lim_{t\to 0}f(t)=\lim_{p\to \infty}pF(p)$$?

Where $\mathcal{L}\{f\}=F$.

 

[mathman]

pF(p) = p∫e^-ptf(t)dt. Integrate by parts with du = pe^-ptdt and v = f(t). Then (assuming f(t) reasonable) let p -> ∞ and you get the desired result.

 

[matematikuvol]

I saw also assymptotics relation
$\lim_{t \to \infty}f(t)=\lim_{p\to 0}pF(p)$
when that relation is valid?

 

[mathman]

I saw also assymptotics relation
$\lim_{t \to \infty}f(t)=\lim_{p\to 0}pF(p)$
when that relation is valid?

I am not familiar with this. However for most cases, both sides = 0.

 

[matematikuvol]

For $1$ both sides are equal $1$. $lim_{t\to \infty}1=1=lim_{p\to 0}p\frac{1}{p}=1$. I think that is correct only if both limits converge.