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|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