Determining the final value of the output of a discrete system
Your final equation is incorrect: it should be

$${\lim_{k\to\infty}f\left(kt\right)=\lim_{z\to1}\left(\left(1-z^{-1}\right)H\left(\text{z}\right)\right)} $$

Where $H(z)= F(z)G(z)$ which is the $\mathcal Z$-transform of the output of the system that has a transfer function of $F(z)$ with $G(z)$ applied to the input.

From that you already have $F(z)$ from the first line, multiply it by $G(z)$ to get the $\mathcal Z$-transform of the function that would be at the output of the system, multiply that product by $(1-z^{-1})$, and then take the limit of the resulting equation as $z$ goes to $1$.