Consider a sample of \(n\) i.i.d. random variables on the real line whose common distribution function \(F\) is regularly varying at infinity with unknown index of variation \(1/r\). A popular estimator of \(r\) is \textit{B. M. Hill's} estimator \(r_ n\) [Ann. Stat. 3, No. 5, 1163-1174 (1975; Zbl 0323.62033)], which is based on the upper \(k_ n\) order statistics in the sample. Under the condition that \(F\) is continuous it is shown, among other things, that, for any \(a\in (0,1)\), \[ \begin{aligned} \lim_{n\to \infty} &k_ n^{-1} \log P(r_ n /r\geq 1+a) =- a+ \log (1+a),\\ \lim_{n\to \infty} &k_ n^{-1} \log P(r_ n/ r\leq 1-a)= a+\log (1-a), \end{aligned} \] if \(k_ n\in \{1,\dots, n\}\) satisfies \(k_ n\to \infty\), \(k_ n/ n\to 0\) as \(n\to\infty\).