Limit of sum is sum of limits $$ \lim_{x\to a } [ f(x) + g(x) ] = \lim_{x\to a } f(x) + \lim_{x\to a } g(x) 
$$
$$ \lim_{x\to a } f(x)   \  \text{ and }  \ \lim_{x\to a } g(x) \ \text{ exist}. $$
I wanted to know that if $f(x)$ and $g(x)$ are defined over an interval $S$,
Does $a$ need to be included in $S$?
 A: Certainly, $\lim_{x\to a} f(x)$ can indeed exist if $a \notin S$.
For example, consider the function $$f(x) =\frac{x^2-1}{x-1}, \text{ defined }\,\forall x, x\neq 1:$$ $$\begin{align}\lim_{x\to 1} \frac {x^2 - 1}{x - 1} 
&= \lim_{x\to 1} \frac{(x-1)(x+1)}{x-1}\\ \\
& = \lim_{x\to 1} x+1 = 2\end{align}$$
Note, the limit as $x$ approaches 1exists, even though the function is undefined at $x = 1$ (i.e. even though $1$ is not in the domain of $f$).
A: It seems like you want to know whether it makes sense to talk about
$$
\lim_{x \to a} f(x)
$$
when $a$ is not in the domain of $f$.
It does make sense. Consider for example the limit
$$
\lim_{x\to 0} \frac{x^2 + x}{x} = 1.
$$
A: 
I wanted to know that if $f(x)$ and $g(x)$ are defined over an interval $S$, Does $a$ need to be included in $S$?

Not necessarily, to put it simply that's because when we use limits, we want to understand the behavior of a function as it approaches a value, and not what the function evaluates to at that particular value. So it doesn't matter at all that the function be defined over that particular value.
A: $\lim_{x \rightarrow a}$ makes sense if and only if $a$ is a limit point of $S$, which is true if and only if $a$ is in the closure of $S$ and is not an isolated point of $S$.
Another way of expressing this is that $a$ is a limit point of $S$ if and only if $S$ contains a sequence of points distinct from $a$ but which converge to $a$.
