Putnam 1965 B4

B-4. Consider the function $$f(x, n)=\frac{\binom{n}{0}+\binom{n}{2} x+\binom{n}{4} x^{2}+\cdots}{\binom{n}{1}+\binom{n}{3} x+\binom{n}{5} x^{2}+\cdots}$$ where $n$ is a positive integer. Express $f(x, n+1)$ rationally in terms of $f(x, n)$ and $x$. Hence, or otherwise, evaluate $\lim _{n \rightarrow \infty} f(x, n)$ for suitable fixed values of $x$. (The symbols $\binom{n}{r}$ represent the binomial coefficients.)