Is there integer functions $f(n)$ and $g(n)$, such that $\lim_{n \to \infty} f(n)\zeta(2) + g(n) = 0$ where $f(n)\zeta(2) + g(n) \neq 0$ for all positive integer $n$.
For example:
$(-1)^n!n e + (-1)^{n+1}n! \neq 0$ for all positive integer $n$ and the limit at infinity is $0$. Here $f(n) = (-1)^n!n$ and $g(n) = (-1)^{n+1}n!$ are integers functions for all positive integer $n$. Instead of $e$ I want to use $\zeta(2)$.
For any real number $x$, there exist functions $f,g$ so that
$$\lim_{n\to\infty} xf(n)+g(n)=0.$$
If $x$ is rational, then this is trivial. Otherwise, let $\frac{p_n}{q_n}$ be the $n$th convergent to the continued fraction of $x$, and let $g(n)=-p_n$, $f(n)=q_n$.
For more about continued fraction convergents, see Wikipedia.