On existence of PI-exponents of unital algebras

We construct a family of unital non-associative algebras ${T_\alpha\vert~ 2<\alpha\in\mathbb R}$suchthat$\underline{exp}(T_\alpha)=2$,whereas$\alpha\le\overline{exp}(T_\alpha)\le\alpha+1$. In particular, it follows that ordinary PI-exponent of codimension growth of algebra $T_{\alpha }$ does not exist for any $\alpha >2$. This is the first example of a unital algebra whose PI-exponent does not exist.