If a first derivative doesn't exist at a certain point, is it not a critical point? Say $f'(x)$ of a function was $(x+2) \over (x+3)$. If $x = -2$ then $f'(x)$ = 0, which means that at this point, there would be a local min or max. But what if $x = -3$? It doesn't exist for $f'(x)$, so do we just ignore it? State it DNE at $x = -3$ so it is not a critical point?

Quoting Wikipedia :

In mathematics, a critical point or stationary point of a differentiable function of a real or complex variable is any value in its domain where its derivative is $0$. Some authors also classify as critical points any limit points where the function may be prolongated by continuity or where the derivative is not defined.

The above definition can be found here.
Thus, it all really comes down to what you want to define as a critical point. The most standard definition (used in dynamical systems for example) is in the case of the derivative being equal to zero. If that's the case, no, it's not a critical point.
On the other hand, if you want to include to the definition of a critical point any points where the derivative is not defined, then yes, $x=-3$ is a critical point.
