What does the phrase "except possibly at $a$ itself" mean in the definition of a limit? The definition of limit says that Let $f(x)$ be a function defined on some open interval that contains the number $a$, except possibly at $a$ itself. Then we say that the limit of $f(x)$ as $x$ approaches $a$ is $L$ If....{the rest of definition is left to make the question easier}. 
What does the phrase "except possibly at $a$ itself" mean?
What is the significance of defining the interval, open i.e why not closed?
 A: More generally the definition of limit of a function $f$ at a point $a$ requires that $a$ is a limit point for the domain of $f$; see for example Rudin's book third edition definition 4.1.
When $a$ is a limit point for the domain of a function $f$, you are sure that it's always possible evaluate $f(x)$ when $x\rightarrow a$; this make well defined the expression:
\begin{equation}
\lim_{x\rightarrow a} f(x)
\end{equation}
Now remember that if $a$ is a limit point of a set $D$, it may be that $a\in D$ or not.
Many books choose to define limit for functions without the concept of a limit point of a set; so they write conditions to bypass the gap. When your book says "except possibly at $a$ itself", it simply reflects the property of a limit point of a set to belong or not to the set itself.
For the second answer: your book choose the point $a$ in an open interval in order to make possible $x\rightarrow a$ from left or right.
If you choose $a$ to be in a closed interval, it may be possible that $a$ is an end point; for example $a\in [a, b]$; but in this case you can make $x\rightarrow a$ only from right and not from left; this make the definition of "right limit" and not the definiton of limit in general.
Hope this help.
A: Here's an answer at more of a calculus level and not so much a real analysis level.

Let $f(x)$ be a function defined on some open interval that contains the number $a$, except possibly at $a$ itself.

"except possibly at $a$ itself" means the function may not actually be defined at $x = a$.  And a function need not even be defined at $x=a$ in order for the limit as $x \to a$ to exist.  This is because the limit of $f(x)$ as $x \to a$ describes how $f(x)$ behaves near $x=a$ and not necessarily at $x = a$.
For example, $f(x) = \frac{x^2-4}{x-2}$ is not defined for $x=2$.  But we can calculate the limit as follows:
$$\lim_{x \to 2} f(x) = \lim_{x \to 2} \frac{x^2-4}{x-2} = \lim_{x \to 2} \frac{(x-2)(x+2)}{x-2} = \lim_{x\to 2} (x+2) = 4$$
This means that as $x \to 2$, our function $f(x)$ looks (behaves) like it's approaching the value $f(x) = 4$.  And again, the function isn't even defined at $x = 2$, but this is irrelevant for the limit.
To really drive home the point that the value of the function at $x=a$ is irrelevant to the limit as $x \to a$, consider the following three functions:


*

*$f(x) = \frac{x^2-4}{x-2}$

*$g(x) = x^2$

*$h(x) = \begin{cases} x+2, & x \ne 2 \\ 10, & x = 2\end{cases}$


All three of these functions have the same limit as $x \to 2$.  The limit is $4$.  But note the following:


*

*$f(x)$ is not even defined at $x=2$.

*$g(x)$ is defined at $x=2$ and $g(2) = 4$, which is the same value as the limit.

*$h(x)$ is defined at $x=2$ but $h(2) = 10$, which is not the same value as the limit.

A: An open interval that contains $a$ also contains all points sufficiently close to $a$.  That is why an open interval is used in the definition.
"except possibly at $a$ itself" was included because the limit of the function at $a$ is determined by the behavior of the function near $a$ but not at $a$.
