  We study three notions of shadowing: classical shadowing, limit (or
asymptotic) shadowing, and s-limit shadowing. We show that classical and
s-limit shadowing coincide for tent maps and, more generally, for piecewise
linear interval maps with constant slopes, and are further equivalent to the
linking property introduced by Chen in 1991.
  We also construct a system which exhibits shadowing but not limit shadowing,
and we study how shadowing properties transfer to maximal transitive subsystems
and inverse limits (sometimes called natural extensions).
  Where practicable, we show that our results are best possible by means of
examples.
