  We define and discuss transfinite asymptotic notions of smoothability, type,
and equal norm type. We prove distinctness of these notions for a proper class
of ordinals and that each class is an ideal. We also extend some results of
Godefroy, Kalton, and Lancien to operators and ordinals greater than zero
regarding the equivalence of equal norm asymptotic type and uniform renormings
with power type smoothness. Finally, we discuss an extension of a non-linear
result for quasi-reflexive, asymptotically $p$-smoothable Banach spaces to
quasi-reflexive Banach spaces with asymptotic equal norm type $p$.
