# When should the limit be introduced?

Usually school children are taught fractions and decimal representations way before the notion of limit. So they must come to the idea that infinite decimal sequences like 0.999... are the same as an infinite sequence of partial sums 0.9 + 0.09 + 0.0009 + ... And when they are told that 0.999... = 1 they unavoidably get the wrong impression that the sequence has a fixed value.

I know from my own bad experience that many students are not aware of the fact that a strictly increasing sequence has no fixed numerical value since it increases without end. Usually mathematicians know that the limit has to be applied but from time to time I have even heard mathematicians claim: Every finite sequence fails but the infinite sequence is 1 - as if the infinite sequence of partial sums could contain an infinite partial sum or could somehow reach 1! Therefore it is very important to clear up these things early enough in order to prevent such mistakes.

My question: When is the appropriate time to correct this wrong picture and to tell them the truth about the meaning of the decimal representation?