What does the definition of the limit say? I have been having trouble for some days now internalizing the concept of a limit especially as it appears in different contexts as in continuity,derivatives, integrals sequences etc.
Although I understand the definition of the limit in each of these instances, it just doesn't seem to be intuitively obvious, but I believe that I have zeroed in one where the problem lies.
Consider the following two statements:
We say that $\lim_{x\to c}f(x)=L$ if $f(x)$ gets closer to L as $x$ gets closer to c. $(1)$
We can make $f(x)$ arbitrarily closer to L by making $x$ sufficiently close to $c$. $(2)$
Can somenone please explain why $(1)$ and $(2)$ are equivalent?
because is seems that  the  $\epsilon-\delta$ definition addresses $(2)$ not $(1)$  but surely if the definition is correct it must encapsulate the intuitive although vague idea of $(1)$.
 A: For me, (1) is not quite true, if the adjective closer is strictly what "closer" mean. Consider $f(x)=x\sin\frac{1}{x}$, the limit of $f$ at $x=0$ is $0$, but when $x=0.01$, $f(0.01)=-0.00506366$, then when $x$ truly getting more closer to $x=0$, say $x=0.009$, then $f(0.009)=-0.00823449$, the function value leaves the $0$ more, right? So it depends on what you think the English adjective "closer" mean. If you think it mean its strictly meaning, then (1) is not a correct (equivalent) statement of limit.

A: Probably, a sensible thing to say here is this tale of Alice, Bob and Charlie.

Once upon a time, Alice read the $\varepsilon$-$\delta$ definition of limit. In order to revise her study, she had Bob help her by listening to her exposition of the concepts she had learnt. Thus phrase $(2)$ came to be.
  Later on, Bob tried to explain to his curious friend Charlie what they were talking about. Bob not being too subtle with either words or symbols, all he could tell Charlie was $(1)$.

By this I mean that (1) is valid just as long as you interpret (though they should technically mean something else) those "closer" as a poorly stated version of what (2) says, which on the other hand is an acceptable translation in English of the $\varepsilon$-$\delta$ definition.
A: Definition (1) is completely true and in fact was the original definition of Cauchy when he spoke of continuity.  It is formalized by saying that when $x$ is infinitely close but not equal to $c$, the value $f(x)$ is infinitely close to $L$.  For details see Elementary Calculus.
