Why do we need $x \neq c$ in $(\epsilon, \delta)$ definition of limits?

Question

We have the following $(\epsilon, \delta)$ definition of limits:

$$ \lim_{x \rightarrow \ c} f(x) = L \; \text{iff} \\ \forall \epsilon > 0 \;\;\; \exists \delta > 0 \;\;\; \forall x \in Dom(f)\;\;\; 0 < |x - c| < \delta \implies |f(x) - L| < \epsilon. $$

I understand everything in the definition except for this part:

$$ \color{red} 0 \color{red} <\; |x - c| < \delta $$

Why do we need explicitly state that $| x - c|$ is greater than zero? Obviously it means that $x$ can not be equal to $c$. But I don't understand what wrong is happening when $x = c$ so we need to explicitly avoid this case in the definition. I know that $f$ might not be defined at $c$ at all but we say in the definition that $x \in Dom(f)$ so in case if $c \not \in Dom(f)$ it will never be equal to $x$. What am I missing here?

Answer 1

Consider $$ f(x) = \begin{cases}x^2, & x \ne 0\\ 1, & x=0 \end{cases} $$

Do you want $\displaystyle \lim_{x\to 0} f(x)$ to exist? If you want it to exist and be zero, you need to rule out $x=c$ in the definition.

Answer 2

Because if $f$ is defined differently at $c$ than its 'limit' (i.e. removable discontinuity), you will get the wrong answer.