I am trying to calculate the following limit:
$$\lim_{n \to X} \frac{1-g-\psi\left(2-n\right)}{\left(1-n\right)!}$$ where $X$ is a positive integer. However, I am stuck. I tried writing the digamma function using harmonic numbers but I could not make them work as the input is negative. I also tried using asymptotic formulae (i.e. $\ln(x)$ for $\psi(x)$) but again this did not work as the argument is negative. How could I proceed?