[Math] the difference between “arbitrarily close” and “sufficiently close” in term of limits

Question

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 ?

Answer 1

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"

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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.