Why is $\infty$ always a singularity of a differential equation?

Question

Consider Legendre's Equation:$\frac{d^2y}{dx^2}-\frac{2x}{1-x^2}\frac{dy}{dx}+\frac{l(l+1)}{1-x^2}y=0$.
It is quite evident that the coefficients of $\frac{dy}{dx}$ and $y$ do not diverge at $x \to \infty$. So infinity is not a singularity of the equation!
However if we substitute $x=\frac{1}{z}$ and express the ODE as $\frac{d^2y}{dz^2}+P(z)\frac{dy}{dz}+Q(z)=0$, we get $P(z)=\frac{2}{z}+\frac{2}{z^3-z}$, which clearly diverges at $z \to 0$.
I don't understand the flaw in either of the two ways but they don't agree in any case.
What's going on?