What does 'computable in the limit from above' mean?
To quote Wikipedia/Scholarpedia, 


  A partial function $f$ from the rational numbers to the real numbers is
  upper semicomputable if it is defined by a rational-valued partial
  computable function $\phi(x,k)$ with $x$ a rational number and $k$ a
  nonnegative integer such that $\phi(x,k+1)≤ϕ(x,k)$ for every $k$ and
  $\lim_{k \to \infty}\phi(x,k)=f(x)$. This means that $f$ can be computably approximated
  from above.


Further, using the squeeze theorem, we could conclude:


  If a function $f$ is both upper semicomputable and lower semicomputable
  on its domain, then we call $f$ computable