how to compute $\lim_{n\to\infty}\ln(f(n))/\ln(n)$ for this given $f(n)$?

Question

Let $c=f(0)$ be a real number and $n,m$ positive integers. Let $P_k(n)$ be integer polynomials of $n$.

Let $f(n)$ be defined by the recursion $\left[\sum_{k=0}^{m} P_k(n)f(n+k)\right]+P_{-1}(n)=0$.

If $$\lim_{n\to\infty}\frac{f(n+1)}{f(n)} = 1$$ How does one compute $\lim_{n\to\infty}$ $\ln(f(n))/\ln(n)$ ?