Limit of a function with an undefined feature How is it possible to define $\lim_{x \rightarrow a} f(x)$ if $f(x)$ is undefined at $a$? There is an infinite amount of $x$-values on either side of $x=a$, and without a boundary or a strategically-placed, removable discontinuity, isn't the limit unknown?
I am using this definition of a limit [1].

A function $f(x)$ approaches a limit $A$ as $x$ approaches $a$ if, and
only if, for each positive number $\epsilon$ there is another,
$\delta$, such
that whenever $0 < |x-a| < \delta $ we have $|f(x) - A|< \epsilon$.
That is, when $x$ is near $a$ (within a distance $\delta$
from it), $f(x)$ is near $A$ (within a distance $\epsilon$ from it).
In symbols we write $\lim_{x \rightarrow a} f(x) = A$.

[1] David V. Widder. Advanced Calculus. Dover 1989.
 A: A limit uses the values of $f(x)$ everywhere except at $x=a$ ! Whether $f(a)$ is defined or not and whether $f(a)$ coincides with the limit or not is irrelevant. This is expressed in the definition by $0<|x-a|$.
The goal of a limit is precisely to "guess" what $f(a)$ is should be.
A: I present you with the grap of the function $\;f(x)=\cfrac{\sin x}x\;$ :

Clearly the functions isn't defined at $\;x=0\;$ , yet the limit if the function when $\;x\to0\;$ is $\;1\;$ .
In words: the limit $\;\lim\limits_{x\to x_0}f(x)\;$ doesn't depend at all on whether $\;f(x_0)\;$ is defined or what its possible value could be, but only depends on the behavior of the function on an open punctured neighborhood of $\;x_0\;$ , meaning: without the point $\;x_0\;$ itself.
A: The limit of $f$ at a point $a$ is defined as
\begin{equation}
\lim_{x\to a}f\left(x\right) = L \longleftrightarrow \forall \epsilon > 0, \exists \delta > 0, \forall x, 0 < \lvert x-a \rvert < \delta \longrightarrow \lvert f\left(x\right) - L \rvert < \epsilon.
\end{equation}
You can see that the condition $0 < \lvert x-a \rvert < \delta$ does not require the definition of $f$ at $a$. Whenever you would like to tell if $L$ exists, you may first consult the definition. If it is not straigtforward to tell from the definition, you may use the Cauchy theorem of limit.
Your question is based on intuitive understanding of math, which is the type of math you learned at high school. Unfortunately, contemporary math is based on bottom-level logical systems. Using rigorous logical language is the gap between math in high school and math in college.
