The repeating nines situation is a great example of where two things might have the same value, but still be different. This type of stuff happens all the time in real life.
For example: I might own a business and render you a service. The service costs a thousand dollars. I prefer cash on the barrel head, but you hand me a credit card. I take the credit card, run the charge and consider that good. Paying me with cash or credit has the same effect. It balances the account between us.
A month later, I get some pathetic late night call from you. Being a nice guy, I rush down to the courthouse and bail out that deadbeat brother of yours. I write a thousand dollar check. You come to my place of business an hour later and hand me your credit card to pay your bill.
This time, however, I am livid. A credit card balanced the account in the past. This time I am yelling at you. What is wrong with me? Didn't I say cash and credit were equivalent? $1000 = $1000.
Our could it be that there is a difference between the two figures? The two numbers might have the same value but still be different. As you probably know, merchants have to pay a service fee to run credit card charges. Business people eat the fee as a cost of doing business, but in a straight cash transfer, there is no margin. Equal values aren't always the same.
I believe that the mystery of calculus is better understood in the old Aristotelian sense of potential infinity. The summation of summation of 9/10 + 9/100 + 9/1000 ... is something different than one...although they have the same value. For that matter, it is probably this having the same value while being different that makes calculus work.