  We show that, also within the class of representation-tame finite dimensional
algebras $\Lambda$, the big left finitistic dimension of $\Lambda$ may be
strictly larger than the little. In fact, the discrepancies $Findim \Lambda -
findim \Lambda$ need not even be bounded for special biserial algebras which
constitute one of the (otherwise) most thoroughly understood classes of tame
algebras.
  More precisely: For every positive integer $r$, we construct a special
biserial algebra $\Lambda$ with the property that $findim \Lambda = r + 1$,
while $Findim \Lambda = 2r + 1$. In particular, there are infinite dimensional
representations of $\Lambda$ which have finite projective dimension, while not
being direct limits of {\it finitely generated\/} representations of finite
projective dimension.
