For those that are interested, the problem has been taken from the Elementary Differential Equations and Boundary Value Problems, 8th Edition, by Boyce. It is problem 1 from Section 5.4.
The problem states to find all singular points of the given equation and determine whether each one is regular or irregular.
The point x = 0 is a singular point.
So all other points are ordinary points.
So as my conclusion the two limits are finite numbers and we can say that the point x = 0 is a regular singular point