I am wondering what the relevance of the "for all $x$" part of the epsilon-delta definition of limit is and whether this would be a different definition if we had only a single $x$ for which if $0<|x-a|<\delta$, then $|f(x)-l|<\varepsilon$– would there be any function which approaches a certain limit at a certain point according to this modified epsilon-delta definition with a single $x$, but which doesn't approach this limit according to the standard epsilon-delta definition? The definition I am referencing is the below one:
The function $f$ approaches the limit $l$ near $a$ means: for every $\varepsilon>0$ there is some $\delta>0$ such that, for all $x$, if $0<|x-a|<\delta$, then $|f(x)-l|<\varepsilon$.
I am wondering this because Spivak's provisional definition of a limit was
The function $f$ approaches the limit $l$ near $a$, if for every number $\varepsilon>0$ we can make $|f(x)-l|<\varepsilon$ by requiring that $|x-a|$ be sufficiently small, and $x \neq a$.
This makes it sound as if a single value for $x$ is necessary to make the distance between $f(x)$ and $l$ within $\varepsilon$, leading me to question whether these are the same, and what the relevance of the 'for all' is in the stand epsilon-delta definition.
EDIT: due to the comments, I now see that Spivak wasn't talking about a single $x$ satisfying that condition; would there be any function which approaches a certain limit at a certain point according to this modified epsilon-delta definition with a single $x$, but which doesn't approach this limit according to the standard epsilon-delta definition, so that I can understand the relevance of the "for all $x$".
Best Answer
Not exactly an answer to the question and similar in spirit to @colt_browning's excellent answer, but here is an example with limits at infinity (which I believe are more intuitive than limits at some number $a$):
Consider $f(x)=\sin(x)$. Then, for every $M>0$, there exists $x>M$ such that $f(x)=0$. But, of course, $\lim_{x\to\infty}\sin(x)\neq 0$.