  We establish error bounds of the Lie-Trotter splitting ($S_1$) and Strang
splitting ($S_2$) for the Dirac equation in the nonrelativistic limit regime in
the absence of external magnetic potentials, with a small parameter
$0<\varepsilon\leq 1$ inversely proportional to the speed of light. In this
limit regime, the solution propagates waves with $O(\varepsilon^2)$ wavelength
in time. Surprisingly, we find out that the splitting methods exhibit
super-resolution, in the sense of breaking the resolution constraint under the
Shannon's sampling theorem, i.e. the methods can capture the solutions
accurately even if the time step size $\tau$ is much larger than the sampled
wavelength at $O(\varepsilon^2)$. $S_1$ shows $1/2$ order convergence uniformly
with respect to $\varepsilon$, by establishing that there are two independent
error bounds $\tau + \varepsilon$ and $\tau + \tau/\varepsilon$. Moreover, if
$\tau$ is non-resonant, i.e. $\tau$ is away from certain region determined by
$\varepsilon$, $S_1$ would yield an improved uniform first order $O(\tau)$
error bound. In addition, we show $S_2$ is uniformly convergent with 1/2 order
rate for general time step size $\tau$ and uniformly convergent with $3/2$
order rate for non-resonant time step size. Finally, numerical examples are
reported to validate our findings.
