What is the proof for the Laplace transform of f(t)/t?

[fishingspree2]

Hello, I am trying to find the laplace transform of \frac{f(t)}{t}

http://nptel.iitm.ac.in/courses/Webcourse-contents/IIT-KANPUR/mathematics-2/node92.html
Lemma 10.3.9

I don't understand why we are taking the limit as s goes to infinity

Can anyone help me?
thank you

 

[LCKurtz]

Look at it this way. You have this relationship:

F(s) = -\frac d {ds} G(s) = -G'(s)

Take the anitiderivative of both sides from s to b

\int_s^b F(u)\, du = -(G(b) - G(s)) = G(s)-G(b)

This must hold for all b, so let b\rightarrow \infty, knowing G(b)\rightarrow 0 and switching sides:

G(s) = \int_s^\infty F(u)\, du = \int_s^\infty F(s)\, ds

This builds in the constant of integration in the formula.