We have the series expansions of digamma and trigamma functions for $x>0$, $$\psi^{(0)}(x)=-\gamma-\frac1x+\sum_{k=1}^{\infty}(\frac1k-\frac1{x+k})$$ and $$\psi^{(1)}(x)=\sum_{k=0}^{\infty}\frac{1}{(x+n)^2}$$ respectively. Looking at the graphs of these functions, I saw a unique intersection point. Graph of digamma function looks like a shifted logarithm function and graph of trigamma looks like an hyperbola. İnspired from this I thought the intersection point must be close to the intersection of $y=\ln x-\gamma$ and $y=\frac1{x^{2}}$ which is $\approx 2.19$ for the $x$-coordinate. Note that the real value is $\approx 2.23$.
Questions: Can my guess be justified? Is there a literature for this intersection point? Thanks in advance.