I suppose is well known to you the so called 'Final value theorem' , connected to the Laplace Trasform:

If...

... then is...

... provided that both limits exist.

Let's take an example and consider the function...

(1)

... where is the so called 'Haeviside step function'. This function is represented here...

In this case...

(2)

... so that is...

(3)

... and it seems to be all right. Let' try now with another function...

(4)

... that is represented here...

It is quite obvious that such type of function doesn't have limit for . In this case is...

(5)

... and if we try to apply the 'Final value theorem', using three times the L'Hopital rule, we obtain...

(6)

... and that's is a little surprising! ...

At this point the questions for You are...

a) did I make some mistrake?...

b) if not how to interpret all that?...

Any help will be gratefully accepted!...

Kind regards