How to identify limit ordinals?

Definition: An ordinal number $\alpha$ is called a limit ordinal number if there is no ordinal number immediately preceding $\alpha$.

Now my lecture notes say that $\omega, 2\omega, \omega^2, \omega^\omega$ are limit ordinal numbers whereas $\omega+3,2^\omega+5$ are not which is intuitively clear. But is there a characterization of limit ordinals that may work when proving the case with some rigor? Or is it "just look and identify" sort of a thing using the ordering of the family of ordinal numbers?

There can be several answers, depending on what you mean by "identify".

The simplest answer would be to look at the Cantor normal form of $\alpha$, and see if it has any finite ordinal there. If the answer is no, then $\alpha$ is a limit ordinal (or $0$, which may or may not be a limit ordinal depending on you convention) and otherwise it is a successor ordinal.

Another answer would be that $\alpha$ is a limit ordinal if and only if for every $\beta<\alpha$, $\beta+1<\alpha$ (with the same caveat about $0$ as before). Although it seems not to be exactly what you are looking for.

You already have a characterization of limit ordinals. In any specific case you just have to verify that this characterization is fulfilled. Consider for example $\omega^2$:

$$ \omega^2 = \omega \cdot \omega = \sup \{ \omega \cdot n \mid n < \omega \}. $$

Hence, if $\alpha < \omega^2$, there is some $n < \omega$ such that $\alpha < \omega \cdot n$. But then $$ \alpha + 1 < \omega \cdot n + 1 < \omega \cdot (n+1) \le \omega^2. $$

Thus $\omega^2$ is a limit ordinal.