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.
Best Answer
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.