Why do we include deleted neighborhoods when defining limits? 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 $x \neq a$, for the case of discontinuity, it is clear. However the book I am reading also doesn't allow $x \neq a$ for continuous functions. 
 A: 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.)
A: 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.
A: Note that LIMIT is a DYNAMIC PROCESS, not STATIC equilibrium. Therefore the value of f precisely at the point a is irrelevant. It is important to understand that the limit of a function is the condition on the behavior
of a function in a deleted neighborhood of a point, rather than a condition on its value at a particular point.
Will be revisited after I find my old book note when I made at the first sight of limit many years ago. 
