Frobenius & Special Functions · Superchallenge problem

Let $y(x) = \sum_{n\ge0}a_nx^n$ be the Frobenius solution, with $a_0 = 1$, of $x(1-x)y'' + (1 - 2x)y' - \tfrac14y = 0$ about its regular singular point $x = 0$. The series converges for $|x| < 1$ and $y(x) \to \infty$ as $x \to 1^-$. Compute $\displaystyle\lim_{x\to1^-}\frac{y(x)}{\ln\frac{1}{1-x}}$.