The definition from my book states the following regarding a limit $L$ of a function $f:\Bbb{R^1}\mapsto \Bbb{R^1}$. "$f$ tends to limit $L$ as $x$ tends to $a$ if and only if i ) there is an open interval $I$ containing $a$ which, except possibly for the point a, is contained in the domain of $f$..."
So before proceeding with a limit proof by definition, we need to always specify some open $I\subset D_f$, such that $f(x)$ is defined at each $x\in I$ ( save for $x=a$ ). Does this mean that, in order to prove a limit exists, it is sufficient to take only the case where $x\in I$?
When you are considering the limit of a function, you need to be able to define the function arbitrarily close to the point at which you want to find the limit.
A useful definition is that a continuous function tends towards $L$ at $x^*$ if its function values $f(x_i)\rightarrow f(x^*)$ when $x_i \rightarrow x^*$. Now note that we need to be able to define these $f(x_i)$, so they must be in the domain of $f$. Also, as $x_i \rightarrow x^*$, infinitely many $x_i$ are in an open set of arbitrarily small size around $x^*$! So you need only worry about those arbitrarily close to $x^*$. If you can define your open interval $I$, you might as well only consider $x$ in $I$.