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 ? 
 A: 
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.

A: *

*arbitrarily close: true for any closeness, how large or how small; $\forall\epsilon$.


$|x-1|>0$ holds for $x$ arbitrarily close to $1$. 


*

*sufficiently close: true under some suitable upper bound; $\forall\epsilon<u$ where $u$ is dictated by the problem on hand.


$|x-1|<1$ holds for $x$ sufficiently close to $1$. 
A: You do not mention where your function $f$ is defined and which range it has, but let us assume that it is a function $f : \mathbb R \to \mathbb R$. Your definition
"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 $a$"
of the limit of $f$ is nothing else than a verbal rephrasing of the standard $\varepsilon$-$\delta$-definition. Arbitraritly close means that for each $\varepsilon > 0$, how big or small it may be, you get
$\lvert f(x) - L  \rvert <  \varepsilon$ by restricting $x$ to be sufficiently close to $a$ but not equal to $a$.
Sufficiently close means that there exists some $\delta > 0$ such that
$\lvert f(x) - L  \rvert <  \varepsilon$ for all $x \ne a$ with $\lvert x - a \rvert < \delta$.
Thus "arbitraritly close to $L$" means "for all neighborhoods of $L$" and "sufficiently close to $a$" means "there exists a neighborhood of $a$".
You see that this is not interchangeable.
