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For $b\in\mathbb{N}, b\ge2$ we determine the limit points of certain subsetsof  $$  \left\{\frac{b^n\pmod{n}}{n}:n\in\mathbb{N}\right\}.  $$ As a consequence, we obtain the density of the latter set in $[0,1]$, aresult first established in 2013 by Cilleruelo, Kumchev, Luca, Ru\'{e} andShparlinski..

Some remarks related to the density of $\{(b^n\pmod n)/n:n\in\mathbb{N}\}$