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.