I'm looking for the following limit


$$\lim\_{x\rightarrow 0} \frac{\sqrt{\frac{\text{BesselK}^{(2,0)}(0,x)}{\text{BesselK}(0,x)}}}{\log (x)}$$


I believe the limit is finite, and is near -0.578. However, it is hard to get more digits numerically, and I'm interested in knowing if there is a closed form behind this constant. 


$\text{BesselK}^{(2,0)}$ is the second derivative of the Bessel K function with respect to its first variable.



Limit involving modified Bessel Function of the second kind
Using series representations, it turns out to be $-\frac{1}{\sqrt{3}}$


