Power of a real exponential signal
I think the correct expression for the power is
$$ P = \lim_{T\rightarrow\infty}\frac{e^{-aT}-e^{aT}}{2aT}. $$
When $T \rightarrow \infty$, $e^{aT}$ tends to infinity &quot;faster&quot; than $1/T$ and the limit does not exist.
However, an interesting thing happens when $a=0$. The signal $x(t) = e^{-at}$ becomes $e^0=1$, which is a power signal (with power equal to 1 watt).
I'm sure that (using L'Hopital or some other trick) it can be shown that $P=1$ when $a=0$, but I have not tried it. In any case, I would say that the premise of the question is wrong, and in fact $x(t)$ is a power signal for at least one (real) value of $a$.