How to show that $$ \lim_{x\to0}\frac{\Gamma(x)}{\psi(x)}=-1 $$ where $\psi(x)$ is the digamma function.
2026-04-29 09:41:06.1777455666
Show $\lim_{x\to0}\frac{\Gamma(x)}{\psi(x)}=-1$
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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.