Understanding Final Value Theorem in Laplace Transform: Tips & Examples

[zoom1]

http://fourier.eng.hmc.edu/e102/lectures/Laplace_Transform/node17.html

On that page about final value theorem, it says that;

If there are pairs of complex conjugate poles on the imaginary axis, will contain sinusoidal components and is not defined.

However at the bottom of the page, in order to find the DC gain, it uses Final value theorem.

Ok, well, let's assume somehow he put "0" where he saw "s". How about multiplication with "s" for final value theorem on s domain ?
He didn't even multiplied the H(s) with "s" ?

Confused.

 

[zoom1]

http://fourier.eng.hmc.edu/e102/lectures/Laplace_Transform/node17.html

On that page about final value theorem, it says that;



However at the bottom of the page, in order to find the DC gain, it uses Final value theorem.

Ok, well, let's assume somehow he put "0" where he saw "s". How about multiplication with "s" for final value theorem on s domain ?
He didn't even multiplied the H(s) with "s" ?

Confused.

Seems like I missed the unit step input, so "s" will be canceled by "1/s". However still the first question holds. System has complex conjugates lying on the left side of the s plane. Which makes the output sinusoidal. So, x(infinite) shouldn't be defined.