That famous Cauchy definition of the limit, and another view that explains the elusive concept.

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If you can come arbitrarily close to a value, in the limit process, then that value is the limit. Of course, there has been always a question "but there is still that small error there, no matter how many elements we add, and no matter how close we are". It is true if you do not let n -> infinity. If n-> infinity then error goes to zero. But there is another nice thing about it. The "arbitrary close" statement guarantees that WHEN n -> to infinity that value will be the limit. It does not say it is the limit if you have finite number of values, no matter how big that number is. It says that, essentially, the fact that you can come "arbitrarily close", i.e. "close as much as you want" to that value, in that, and only in that case, it guarantees that, when n-> infinity, that value is the limit. No other statement will guarantee that. No other statement will guarantee that anything similar will happen when n-> infinity. That's the statement you want.

[ applied math, definition of limit, limit in mathematics, concept of a limit, limit, limiting value, integral, differential, ]