According to Abbott 4.2.5 "Divergence Criterion for Functional Limits",
Let $f$ be a function defined on $A$, and $c$ be a limit point of $A$. If there exist two sequences $(x_n)$, $(y_n)$ with $x_n \neq c$ and $y_n \neq c$ and $\lim_{x \to c} x_n = \lim_{x \to c} y_n = c$ but $\lim_{x \to c} f(x_n) \neq \lim_{x \to c} f(y_n)$, then $\lim_{x \to c} f(x)$ does not exist.
Does this hold when $\lim_{x \to c} f(x_n)$ does not exist, and $\lim_{x \to c} f(y_n)$ does not exist?
For example, say $(x_n)$ converges but $f(x_n)$ oscillates.
The limits, at least, would not be equal in this case.
I will leave this question here, cautious, since my previous analysis question was closed for "lack of focus", where a commenter described it as "a word salad". Stack Exchange wants enough but not too much information, it seems.
There are many answers on this site where the limits do exist, but I struggle to find examples of divergence criterion where e.g. $f(x_n)$ oscillates as I stated above.
If you would like more details, please let me know. I am doing a homework problem involving the convergence or divergence of a particular function unstated here.
Yes: In fact , it is not too difficult to prove that $f(x)\rightarrow L$ as $x\rightarrow c$ , if and only if for any sequence $(x_n)$ , such that $x_n\rightarrow c,$ $ x_n\ne c$ (and $x_n$ lie in the domain of f) , $f(x_n)\rightarrow L$ by the definition of convergence of sequences and the definition of a limit of a function.
If the limit doesn't exist for any particular sequence of points: Then the function definitely doesn't converge to a limit.
Doing that proof may be a useful exercise to convince yourself of this fact.