I am trying to wrap my head around the formal definition of a limit using the delta-epsilon approach, which I'll repeat below for convenience:
Suppose that $L$ and $c$ are real numbers and $f$ is a real-valued function defined on some open interval containing $c$. We say that $\displaystyle{\lim_{x \to c} f(x) =L}$ if for every $\epsilon > 0$, there exists a $\delta > 0$ such that whenever $0 < |x - c| < \delta$, then $|f(x) -L| < \epsilon$.
More informally, $|a - b|$ is like saying "the distance between $a$ and $b$", so I'd like to rephrase the definition in my own informal interpretation:
I can give you a positive real $\epsilon$, defining a threshold on the $y$-axis centered on $y=L$, where the threshold is of size $\epsilon$ on the top and bottom. And no matter what value I give you, you can give me back another positive real $\delta$ that describes a similar horizontal threshold on the $x$-axis centered on $x=c$, where the threshold is of size $\delta$ on the left and right.
Moreover, for every $x$ coordinate in this $\delta$ threshold, the $y$ coordinates of the corresponding points in $f$ fall entirely inside the $\epsilon$ threshold.
To my eye this suggests an infinite recursion. Because the threshold that $\delta$ maps to falls within $\epsilon$, whatever that new threshold is would become a new $\epsilon$ that is even smaller, and we'd repeat this process over again, forever.
As $x$ and $c$ get closer and closer, so do $f(x)$ and $L$. Maybe they get closer together at different rates, but as far as I can tell this only says that these two quantities can get closer and closer indefinitely. But what bothers me is that we're saying we can make these closer and closer but say absolutely nothing about what happens at $f(c)$. The limit would be the same even if we removed the point (assuming it was defined). The definition only references what's going on around the point -- the point itself is almost an afterthought.
What I struggle with now is trying to understand how this becomes a useful definition that underlies almost everything in calculus, when the entire concept appears to be about what's happening around something but not on it. What problem is this addressing? What niche is it filling?
We require a punctured neighborhood in the definition of a limit precisely so we can distinguish behavior near a point from behavior at a point.
This distinction is crucial to differential calculus, for example, because we want to define the slope of the tangent line at a point (say $x=a$) to be the limit of the slope of secant line approximations near that point—but the slope of the secant line is
$$\frac{f(x)-f(a)}{x-a}$$
When we let $x$ "approach" or "tend to" $a$, we better not require the $\epsilon$ condition to be satisfied at every point in some interval around $a$, because the above expression isn't defined when $x=a$! So instead we delete the point $x=a$ from consideration.
If our definition required the $\epsilon$ condition to be satisfied at all points in some interval around $a$, limits would exist only for continuous functions—functions for which behavior near a point coincides with behavior at the point.