We examine a specific approximating process for the singular integral $$ S^*(f;x)\equiv\frac1\pi\int_{-1}^{+1}\frac{f(t)}{\sqrt{1-t^2}(t-x)}\,dt\quad(-1<x<1), $$ taken in the principal value sense. We study the influence of some local properties of the function $f$ on the convergence of the approximations. Next, assuming that $S^*(f;c)=\lim\limits_{x\to c}S^*(f;x)$, where $c$ is an arbitrary one of the endpoints $-1$ and $1$, we show that the conditions which guarantee the existence of the limiting values $S^*(f;c)$ ($c=\pm1$) and, moreover, the convergence of the process at an arbitrary point $x\in(-1,1)$ are not always sufficient for convergence of the approximations at the endpoints.