Introduction to Indefinite Integrals - At A Glance:
Limits are something the human race (at least, the mathematicians) have created as a result of trying to find a way to think about infinity.
Another way of thinking about infinity involves things called infinitesimals. An infinitesimal is a number that's bigger than 0, but smaller than all normal everyday fractions. If ε is an infinitestimal, then
0 < ε.
This website has a great discussion of limits and infinitesimals.