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In a recent MNRAS article, Raposo-Pulido and Pelaez (RPP) designed a schemefor obtaining very close seeds for solving the elliptic Kepler Equation withthe classical and the modified Newton-Rapshon methods. This implied animportant reduction in the number of iterations needed to reach a givenaccuracy. However, RPP also made strong claims about the errors of their methodthat are incorrect. In particular, they claim that their accuracy can alwaysreach the level $\sim5\varepsilon$, where $\varepsilon$ is the machine epsilon(e.g. $\varepsilon=2.2\times10^{-16} $ in double precision), and that thisresult is attained for all values of the eccentricity $e<1$ and the meananomaly $M\in[0,\pi]$, including for $e$ and $M$ that are arbitrarily close to$1$ and $0$, respectively. However, we demonstrate both numerically andanalytically that any implementation of the classical or modifiedNewton-Raphson methods for Kepler's equation, including those described by RPP,have a limiting accuracy of the order $\sim\varepsilon/\sqrt{2(1-e)}$.Therefore the errors of these implementations diverge in the limit $e\to1$, anddiffer dramatically from the incorrect results given by RPP. Despite theseshortcomings, the RPP method can provide a very efficient option for reachingsuch limiting accuracy. We also provide a limit that is valid for the accuracyof any algorithm for solving Kepler equation, including schemes like bisectionthat do not use derivatives. Moreover, similar results are also demonstratedfor the hyperbolic Kepler Equation. The methods described in this work canprovide guidelines for designing more accurate solutions of the elliptic andhyperbolic Kepler equations.

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