What is the difference between "arbitrarily close" and "sufficiently close" in term of limits? The definition of limit as always > “the limit of $f(x)$, as $x$ approaches $a$, equals $L$” means we can make the values of $f(x)$ arbitrarily close to $L$ by restricting x to be sufficiently close to $a$ but not equal to $a$. What exactly mean by phrases "arbitrarily close" for $f(x)$ and "sufficiently close" for $x$ ? Are they interchangeable ?

> the limit of $f(x)$, as $x$ approaches $a$, equals $L$” means we can make the values of $f(x)$ **arbitrarily close** to $L$

Rephrasing: " _as close as we want_ "

> by restricting x to be **sufficiently close** to $a$ but not equal to $a$.

Rephrasing: " _close enough_ "

* * *

Or put differently: $L$ is the limit (as $x$ approaches $a$) if we can

* make the _distance_ from $f(x)$ to $L$ **as small as we want** ,
* by only making the _distance_ from $x$ to $a$ **small enough**.