a limit of the digamma function

Question

Numerical calculations show that:

$$ \lim_{x\to 0^+}\left(\frac{1}{x}-\psi(x)+\psi(x/2)\right) = 0.$$

Can anybody help to find a formal proof of the above limit?

Answer 1

For $\psi ^{(0)}(x)$, Taylor expansion around $x=0$ is given by $$-\frac{1}{x}-\gamma +\frac{\pi ^2 x}{6}+\frac{x^2 \psi ^{(2)}(1)}{2}+\frac{\pi ^4 x^3}{90}+\frac{x^4 \psi ^{(4)}(1)}{24}+\frac{\pi ^6 x^5}{945}+O\left(x^6\right)$$ Apply it to the expression and the final result is $$-\frac{\pi ^2 x}{12}$$