Very often, we define the limit of a function as $0 < |x -a|< \delta \implies |f(x) - L|< \epsilon$.
A lot of times we don't let $x$ equal $a$, and the reason for this is clear in the case of discontinuity. However, the book I am reading also doesn't allow $x = a$ for continuous functions.
On
While I agree that you could let $x=a$ in some cases, forbidding this reinforces the idea that a limit is a property that holds near a point, and is defined in these terms: i.e. a limit $L$ exists if it is the unique number (real or complex depending upon the context) that is arbitrarily close to $f(x)$ whenever $x$ is arbitrarily close to $a$ but not equal. We define the number $L$ to be the limit if it has precisely this property. Deleting $a$ does not change the limit if it exists.
The function may be undefined at $x=a$. For example, take $a=0$ and let $f(x)=(\sin x)/x\;(x\in \Bbb R;x\neq 0).$ Then the limit is well defined at $0$, even though $f$ is not. (Of course, in this case, we can easily and naturally extend the domain of $f$ by using the limit.)