About Puiseux series of function $\textbf{K}(1-x)$

Question

$\textbf{K}$:Elliptic K Function
I tried to calculate $\lim_{x\to 0}{\frac{\textbf{K}(1-x)}{\ln(x)}}$.
I used Taylor series and L'Hôpital's rule and failed.
I used Mathematica and it gave me Puiseux series of $\textbf{K}(1-x)$: $$\textbf{K}(1-x)=2\ln(2)-\frac12\ln(x)+(-\frac14+\frac12\ln(2)-\frac18\ln(x))x+O(x^2)$$ Wikipedia says Puiseux series are like: $$f(x)=\sum_{k=k_0}^\infty{c_kx^{k/n}}$$ Why does $\ln$ function appear?
And how did Mathematica find the series of it?