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" ?

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.