How Does Corresponding Number Ensure Limit Existence?

[wajed]

If for every number E>0, there exists a corresponding number D>0 such that for all x
|x-x0|<D >> |f(x)-L|<E
Then L is a limit

what is precisely mean of "corresponding number"?
and how can that "correspondance" assure me that the limit exists?

how can a number be corresponding to another number?
I know how can a number be equal/less than/greater than a number, but how can it be corresponding to another number?

I think I understand the definition now, except this "essential?" part

 

[rochfor1]

It just means that if someone tell you E>0, you can tell them D>0 so that the second part is true.

 

[slider142]

Your intention is to foil the following series of arguments by coming up with deltas that work for every scenario:
"You say that f(x) approaches the value L near x=a, but I bet you can't give me an f(x) that is 0.000001 away from L!"
"Of course I can. Just take any x in the interval (a - D, a + D)!"
"Oh okay. but I bet you can't give me an f(x) within 10^-10000 of L!"
"No problem. Let x be in the interval (a - D1, a + D1)!"
"Hmm. What about ..."
and so on. The challenges are the epsilons and the responses are the deltas. You can usually formulate an epsilon-delta argument so that the deltas are some function of the epsilon challenges, or argue by inequalities.