The gist of the epsilon delta definition of limit I am currently re-reading and re-learning with basic concepts in calculus/analysis. I have tried to prove certain limits exist using the epsilon delta definition. Many of these examples are fairly easy because they are made so to guide new students like me to choose the right $\delta$. The gist of these problems is that illustred through this example:
Prove $\displaystyle\lim_{x\to5}|3x+2|=17$.
Let $\epsilon>0, \epsilon$ is arbitrary, pick $\delta=\epsilon/3\implies3\delta=\epsilon$. Then $\forall x\in\mathbb{R}$, then within $0<\color{red}{|x-5|}<\delta$, we have $|3x+2-17|<|3x-15|=3\color{red}{|x-5|}$. This makes $|f(x)-L|$ equal to $|x-a|$, and the $3$, let call it a constant C, they make it easy to choose the value of $\epsilon$ because now you can come to the conclusion that $3|x-5|<3\delta\implies3|x-5|<\epsilon/3\implies|x-5|<\epsilon$.
Stewart does provide a harder problem where such a $C$ is not provided, like $\displaystyle\lim_{x\to3}x^2=9$. I will ask the question on this problem in another thread, but here I wish to ask you something.
I have been spoonfed like many students in Calc 1 with a list of limit law to calculate limits. This means, for example, when x approaches $3$, $x^{2}$ approaches $9$. The term approach seems to be forgotten since for simple limit of polynomial, we can just plug the limit in and it pops out the value.
My point is with the definition of epsilon delta, the gist seems to lie in the concept of neighbourhood or nearness. I have seen the title of a book by Dugac and it contains the word "voisinage" in French. This makes me think that we are interested in the neighbour of the limit point, rather than the exact limit point.
I have also read that the definition of limit is independent from the concept of function being defined. A classic example is that $$\displaystyle\lim_{x\to0}\dfrac{\sin(x)}{x}=1$$, despite the fact that $x$ is not defined at $x=0$. But the limit still exists. There are many proofs to this, most useful is trigonometric proof.
I have looked into cases when limit doesn't exist, and this site suggests that:
https://www.mathwarehouse.com/calculus/limits/how-to-determine-when-limits-do-not-exist.php
The one-sided limits are not equal
The function doesn't approach a finite value (see Basic Definition of Limit).
The function doesn't approach a particular value (oscillation).
The x - value is approaching the endpoint of a closed interval
Of all these, the ones with one-sided limit makes me wonder. Since in one dimension, we can easily define left and right, in 3 dimensions, this may not be very useful, am I correct?
So the point of epsilon and delta is to generalize the concept of neighbourhood or nearness of the limit point, because this can be generalized to higher dimensions?
Sorry, I haven't studied multivariate calculus so my experience lies entirely within function of one variable.
 A: "the point of epsilon and delta is to generalize the concept of neighborhood or nearness of the limit point, because this can be generalized to higher dimensions"
I would say it's not to generalize it but to define it. Usually the delta pertains to $x$, the epsilon pertains to $f(x)$.
In higher dimensions one defines something called an open ball. Well, that is a neighborhood in higher dimensions.
A: Limits are tricky in this sense. It is true that the limit
$$
\lim_{x \to 3}x^2
$$
can be evaluated by simply plugging in $x=3$, but the reason for this is not as simple as you might think. I will explain this in more detail in a moment, but first let's consider what limits are in general. As you have already correctly noted, limits tell us about the behaviour of a function as you approach a value, but not when you actually 'get there'. The example you gave is perfect. When we write
$$
\lim_{x \to 0}\frac{\sin x}{x} = 1
$$
what we mean is that as $x$ approaches $0$, $\frac{\sin x}{x}$ approaches $1$. In more detail, $\frac{\sin x}{x}$ gets arbitrarily close to $1$. Pick a tiny number, say $0.000001$. Provided that $x$ is sufficiently close to $0$, the difference between $\frac{\sin x}{x}$ and $1$ will be smaller than that tiny number. This helps us state the precise definition of limit:

We say that the limit as $x$ approaches $0$ of $\frac{\sin x}{x}$ equals $1$ because we can make $\frac{\sin x}{x}$ arbitrarily close to $1$ by requiring that $x$ is sufficiently close to, but unequal to, $0$.

Hopefully, this is the gist of the definition of a limit that you were looking for. And the precise definition definition with epsilon's and delta's is saying essentially the same thing. It just clarifies our wordy description even further.
Note that the last section of our wordy definition was

...requiring that $x$ is sufficiently close to, but unequal to, $0$.

As we mentioned earlier, limits tell us about what happens we get close to, but never actually reach a certain value. In a way, the limit tells us what we anticipate when we get close to a value, rather what actually happens there. The function $\sin x / x$ is undefined when $x=0$. It makes no sense to actually ask what happens when we 'get to zero'. But we can make meaningful statements about what happens when we get close to zero. Let's go back to your example,
$$
\lim_{x \to 3}x^2 \, .
$$
If limits are all about what happen when we get close to $3$, then why can we simply plug in $x=3$? This is because the function $x^2$ (and all polynomial functions in general) are continuous. A function is said to be continuous at the point $x=a$ if
$$
\lim_{x \to a}f(x) = f(a) \, .
$$
In other words, the anticipated value is the same as the value actually obtained by plugging in $x=a$. Many functions do not have this property. In the earlier example,
$$
\frac{\sin x}{x}
$$
is not even defined at $x=0$, and so it doesn't even make sense to ask whether it is continuous there. Nevertheless, it can be shown that all polynomial functions are continuous everywhere, and so evaluating the limit
$$
\lim_{x \to a}P(x)
$$
where $P$ is an arbitrary polynomial, simply amounts to plugging in $x=a$. This is what is so confusing to newcomers to calculus. Although limits tell you about the behaviour of a function as you get close to a point, a lot of the time you can take a shortcut and just plug in the value being approached. This is because, with a little work, we can show that the 'elementary functions' (polynomials, the exponential function, sine and cosine, etc.) are continuous at the point $x=a$ if they are defined there. But don't let that mislead you. When we say that $\lim_{x \to a}f(x) = L$, we are still talking about the behaviour of a function as you get close to, but unequal to $a$; however, in some instances it is possible to work out this limit simply by plugging, provided that you know beforehand that the function is continuous.
