The limitation of derivation of modified Bessel function of second kind

Question

The final result I draw is related to the integral of modified Bessel function of the second kind. But I can not solve it, and I need a explicit solution Are you willing to help me? Thank all

$I = \mathop {\lim }\limits_{x \to 0} \frac{\partial }{{\partial x}}\frac{{{K_0}(a\sqrt x )}}{{{K_0}(b\sqrt x )}}$ which a and b are real numbers

Answer 1

Since $K_0(a\sqrt{x})\rightarrow-\frac{1}{2}\ln x-\gamma_{\rm Euler}+\ln(2/a)$ for small $x$, one has $$\lim_{x\rightarrow 0}\; (x\ln^2 x)\frac{\partial }{{\partial x}}\frac{{{K_0}(a\sqrt x )}}{{{K_0}(b\sqrt x )}}=2\ln(b/a).$$ The limit $I$ in the OP is divergent.