  We present limits on sterile neutrino mixing using 4,438 live-days of
atmospheric neutrino data from the Super-Kamiokande experiment. We search for
fast oscillations driven by an eV$^2$-scale mass splitting and for oscillations
into sterile neutrinos instead of tau neutrinos at the atmospheric mass
splitting. When performing both these searches we assume that the sterile mass
splitting is large, allowing $\sin^2(\Delta m^2 L/4E)$ to be approximated as
$0.5$, and we assume that there is no mixing between electron neutrinos and
sterile neutrinos ($|U_{e4}|^2 = 0$). No evidence of sterile oscillations is
seen and we limit $|U_{\mu4}|^2$ to less than 0.041 and $|U_{\tau4}|^2$ to less
than 0.18 for $\Delta m^2 > 0.8$ eV$^2$ at the 90% C.L. in a 3+1 framework. The
approximations that can be made with atmospheric neutrinos allow these limits
to be easily applied to 3+N models, and we provide our results in a generic
format to allow comparisons with other sterile neutrino models.
