Convergence of Sequences: Confusion in definition

The basic issue is with finite vs. infinite # of such exception values. With any finite number, there will always be a maximum value among this set, call it $M$. Thus, for any $N \gt M$, it'll then be true that the inequality of $|x_n - L| \lt \epsilon$ holds for all $n$ $\ge N$.

However, if there's an infinite # of such exception values, then you can't really say that the limit of $x_n$ is $L$ since it doesn't consistently "approach" the value, i.e., there's some $\epsilon \gt 0$ such that, no matter how large your $N$ is, there will always be an infinite number of $n \ge N$ such that $|x_n - L| \ge \epsilon$. This goes counter to the basic concept of what a "limit" is supposed to mean, e.g., it doesn't "converge" as stated in the question comment by coffeemath.