This is about laplace transform

Question

how to deduce that

lim_s=0 s u(s) = lim_t=infinite U(t)

 

sorry the equal sign (subscript) should be replaced with an arrow which indicates "approaching", I couldn't find the suitable symbol for that here.

 

 

Answer 1

Proving this presupposes that lim_{t -> ∞}  U(t)   exists.    Since it exits, call it Q

 

Then consider the left hand side 

 

  lim_{s -> 0}  s ∫ U(t) exp(- st) 

 

   Interchanging the operation of taking the limit and performing the integration gives

 

   s ∫  Q exp(-st)  dt =   s Q ∫ exp(-s t) dt  =  s Q (1/s) exp(-s t) |⁰ _∞  = s Q /s = Q

 

Since Q =   limt -> ∞ U(t)     This proves what was desired.