A context$-$free grammar $($V$,\backslash$Sigma $,$ R$,$ S$)$ is $\backslash$epsi $-$free if:

$$ There is at most one rule whose right$-$hand side is $\backslash$epsi $,$ and that is: S $->\backslash$epsi $($here$,$ S is the start symbol$).$

$$ If the grammar contains the rule S $->\backslash$epsi $,$ then S does not appear on the right$-$hand side of any rule.

Show that, for any given $\backslash$epsi $-$free CFG G$,$ there is a size l where if G generates a string using more than l derivation

steps, then L$($G$)$ is infinite. Give an explicit formula for l in your proof $($basically$,$ l must be a computable

function of the stuff in G$),$ and prove that the formula you give is correct.

Also show that this is not true for context$-$free grammars in general.