does $\lim_{x \to x_0} f(x)$ exist when $f(x)$ not defined for some $x$ from neighbourhood

Question

The question: Does $\lim_{x \to x_0} f(x)$ ($\infty$ allowed for $x_0$) exist in case:

1. For any interval containing $x_0$ there are $x$ where $f(x)$ defined (besides $x_0$)
2. For any interval containing $x_0$ there are $x$ where $f(x)$ is not defined. (Besides $x_0$)
3. Limit exists in a sense of $x$ approaching $x_0$ in domain of $f$

This question sounds very simple, but I found it caused so many argues in the community, it just confused me who is right, who is wrong. I encourage to check those links below prior to answering.

Two questions in this site share the same pattern ($f(x)$ not defined everywhere), yet accepted and most upvoted answers given by very high-rated OPs are completely opposite (limit exists, limit doesn’t exist)

1. What is $\lim_{x \to 0}\frac{\sin(\frac 1x)}{\sin (\frac 1 x)}$ ? Does it exist? Approved answer: limit exists

2. Find $\lim_{x \to \infty} (\frac{1}{e} - \frac{x}{x+1})^{x}$ Approved answer :limit doesn’t exist! (ironically there is "limit exisits" answer that is downvoted)

One of the OPs refers to “Rudin’s Principles of Mathematical Analysis”:

$$(\lim_{ \to \infty} f(x) = L) ⟺ (∀>0∃:∀∈>⟹|()−|<)$$ where $D$ is the domain of $f(x)$

That explicitly mentions domain. I wonder if Rudin is wrong in a sense that he gives slightly different definition than others?! (Forgive me questioning Rudin’s authority!)

Answer 1

The issue is that there are different definitions of the concept of limit. Introductory calculus textbooks and courses use a more restrictive one because I suppose the authors want to simplify the material as much as possible. Basically, that definition requires of $f(x)$ to be defined on an (sufficiently small) open interval around both sides of $c$ when we're taking limits as $x\to c$. So, $\lim_{x\to 0}\sqrt{x}$ doesn't exist because $\sqrt{x}$ is never defined on $(-\epsilon, 0)$. And those limits you linked to also don't exist under this definition. You may guess that some people find it too restrictive to be useful.

A more general definition only imposes the requirement that $c$ be a cluster point for the domain $D$ of $f(x)$, and then inserts $x\in D$ to exclude the points where $f(x)$ isn't defined, i.e., the one in Rudin's book. So, now $\lim_{x\to 0}\sqrt{x}$ (and many more limits) makes sense. Unfortunately, the former definition is popular enough and you may well have people insist it's the only correct one. Which one you should be using probably depends on your context: are you a professional mathematician, a student with a professor who has already given their definition, etc.