Clearly, $0\le f\le1$. Let $h:=1-f$, so that $0\le h\le1$, $h(a,0)=0$, and $h(0,b)=1$ if $b\ge1$. Here and in what follows, $a$ and $b$ are nonnegative integers.
We want to show that $h(1,n)\to0$ (as $n\to\infty$).
Let $$g(a,b):=1(a=0)+\frac1{\ln(1+a+2b)}\,1(a\ge1).$$ It is enough to show that $$h(a,b)\le g(a,b) \tag{1}\label{1}$$ for all $a,b$.
Let us do this by induction on $m:=m(a,b):=a+2b$. If $m=0$, then \eqref{1} is trivial. Also, \eqref{1} is trivial if $a=0$ or $b=0$. If now $a\ge1$ and $b\ge1$, then \eqref{1} follows by induction from the given recurrence for $f$ (and the same recurrence for $h$) and the inequality $$g(a,b)\le\frac a{a+b}\, g(a-1,b)+ \frac b{a+b}\,g(a+1,b-1)$$ for $a,b\ge1$. (To use the induction, note that $m(a-1,b)=m(a+1,b-1)=a+2b-1<m(a,b)$.)