TITLE: Show $\lim_{x\to0}\frac{\Gamma(x)}{\psi(x)}=-1$
QUESTION [5 upvotes]: How to show that
$$
\lim_{x\to0}\frac{\Gamma(x)}{\psi(x)}=-1
$$
where $\psi(x)$ is the digamma function.

REPLY [9 votes]: Simply use recurrence relations:
$$
  \frac{\Gamma(x)}{\psi(x)} = \frac{x \cdot \Gamma(x)}{x \cdot \psi(x)} = \frac{\Gamma(x+1)}{x \cdot \psi(x+1) - 1}
$$
Since $\Gamma(1) = 1$ and $\psi(1)$ is finite, the limit readily follows.