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I'm using the Proofwiki definition of limit:

Let $f:(a\,..b) \setminus{\{c}\} \rightarrow \mathbb R$

Let the domain of the function be called $D$.

Then if there is such a number $L$ that

$\forall\epsilon \in \mathbb R_{>0}:\exists\delta \in \mathbb R_{>0}:\forall x \in D:0<|c-x|<\delta \implies |f(x)-L|<\epsilon$

then this number is called limit.

Why can't $c \in D$?

Why is there a rule that $0<|c-x|$ rather than $0 \le |c-x|$?

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    $c$ may be a point of domain of function, but it is not necessary to be so. What is necessary is that $c$ must be an accumulation point of domain of function. Moreover even if $c$ is in the domain of the function, the value of function at $c$ has no role at all in the definition of limit of function as $x\to c$. In other words $x\to c$ explicitly forbids $x=c$. – 2017-02-19

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Once one has a rigorous definition of the limit of a function, one can give a rigorous definition of continuity of a function at a given point, and it is one of the main reasons why the definition of the limit of a function is as we know it. Say we wish to explain why the function $$ f(x)= \begin{cases} x^2, &\text{ if } x \ne 2,\\ 2017, &\text{ if } x =2 \end{cases} $$ is so obviously discountionuous at $c=2.$ It is easy: $$ 2017=\boxed{f(2) \ne \lim_{x \to 2} f(x)}=\lim_{x \to 2} x^2=4, $$ where the penultimate equality is justified by the fact that $f(x)=x^2$ whenever $x \ne 2.$

So the definition of the limit of a function at an inner point $c \in (a,b)$ $$ \lim_{x \to c} f(x)=L: $$

for every $\varepsilon > 0$ there exists $\delta > 0$ such that for all $x \in (a,b)$ with $0 < |x-c| < \delta$ implies $|f(x)-L| < \varepsilon.$

Thus we exclude $c$ from points at which we analyze the $\varepsilon-\delta$ condition above. Moreover, the function may simply not be defined at $c,$ but we can be interested where the values $f(x)$ accumulate as $x$ approaches $c,$ and so on.

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The notion of limit is especially useful exactly when the function is not defined in a point that is an accumulation point of its domain.

In this case the value of the limit says us what is the ''behavior'' of the function near the point.

This is because , in the definition, we assume that $c$ can , in general, not be a point of the domain. And this implies that $x$ cannot assume the value $c$ so $|x-c|$ cannot be nul.

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    Why isn't the case $c \in D$ considered to be of any value as a limit though? Why isn't a separate definition, just for this case, used? – 2017-02-19
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    @asdasdfsss Because it is not wishable to have two different definitions for two different cases when it is posible to have just one that apply always. Note that you can apply that definition even though c is a point of your domain – 2017-02-19
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    The cited definition explicitly say: Let$ f:A_1→A_2$ be a mapping from $A_1$ to $A_2$ defined everywhere on $A_1$ **except possibly** at $c$.... ( https://proofwiki.org/wiki/Definition:Limit_of_Function#Limit_of_Function_on_Metric_Space) – 2017-02-19
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    So $c$ can be in the domain, but it is important that also it can be a limit ( accumulation) point not in the domain. – 2017-02-19
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Excluding $c$ of the domain makes the definition more general, while including it doesn't add anything (since it's obvious $x=c \implies f(x)=f(c)$). Consider the function $\frac{1}{x}$, it is not defined at zero, so we want a definition of limit which allows us to compute the limit as $x$ goes to zero, even though the function is not defined at that point.